Minimal-Area Metrics on the Swiss Cross and Punctured Torus

Abstract

The closed string theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length greater than or equal to an arbitrary constant that can be set to one. Through every point in such a metric there is a geodesic that saturates the length condition, and the saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the “Swiss cross” challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and of negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.

Numerical Implementation

In this “Appendix” we give some details of the numerical solution of the primal and dual programs for the Swiss cross and punctured torus described in Sects. 4 and 6.1 respectively. For both programs, the fundamental domain was discretized using a lattice, and the derivatives and integrals appearing in the constraints and objectives were replaced by lattice derivatives and integrals. Lowest-order derivatives and integrals were employed, both for simplicity and because—as can be seen in the results presented in Sects. 5.1 and 6.2—the functions being approximated are not necessarily smooth. The discretization scheme in shown in detail in subsection A.1 for the illustrative case of the primal program for the Swiss cross in the L frame; the discretizations for the pentagon frame, the dual program, and the punctured torus are closely analogous.

The minimization and maximization in the two programs respectively was carried out in Mathematica using the built-in functions FindMinimum and FindMaximum. The primal program is a constrained minimization problem, which Mathematica solved with an interior point method. The dual program is an unconstrained maximization problem, which Mathematica solved using a quasi-Newton method with Broyden-Fletcher-Goldfarb-Shanno updates.

In subsection A.2, we compare the results of the primal and dual at various resolutions, focusing on the case of the Swiss cross in the L frame, showing that the primal and dual agree to within numerical errors and show good convergence as the resolution is increased, both in the value of the extremal area and in the form of the optimal metric. Similar convergence behavior was observed for the punctured torus.

1.1 Discretization

Here we describe in detail the discretization employed for the primal program on the Swiss cross in the L frame, a program we discussed in Sect. 4.1. The numerical solution requires discretization of the region identified as a fundamental domain in Fig. 10. We explain the discretization using Fig. 38. The square region $0 \le x, y \le 1/2$ has $N^2$ plaquettes, with N some integer. Each plaquette is a little square with edge $\Delta$ given by

$$\begin{aligned} \Delta = {1\over 2N}. \end{aligned}$$

(A.1)

We introduce a (rational) number k such that $kN\in \mathbb {Z}$ is the number of plaquettes that fit horizontally (or vertically). Since the distance from the origin to the vertical edge is $\ell _s/2$ and the edge of a plaquette is $\Delta$ we have

$$\begin{aligned} \tfrac{1}{2} \ell _s \, = \, kN \Delta = \tfrac{1}{2} k \quad \rightarrow \quad k = \ell _s \ \ (kN \in \mathbb {Z}). \end{aligned}$$

(A.2)

This allows us to do numerical work with rational values of $\ell _s$. For $\ell _s= p/q$, with p and q relatively prime integers, we can work with values of N that are arbitrary multiples of q. The larger the value of N the better the accuracy.

Fig. 38

Discretization of the fundamental domain for the primal program applied to the torus with a boundary. The functions $\phi ^1$ and $\phi ^2$ entering the calibrations are defined on the lattice points. Derivatives and the metric are defined at the center of the plaquettes

Each lattice point is labeled as (i, j) where i and j are positive integers. Notice that the origin is chosen to be the point (1, 1). The (x, y) coordinates of the point (i, j) are

$$\begin{aligned} (x[i], \, y[i]) = \Delta \bigl ( i-1, \, j-1 \bigr ) = {1\over 2N} \bigl (\, i-1, \, j-1 \bigr ). \end{aligned}$$

(A.3)

The function $\phi ^1(x,y)$ is represented by its values $\phi [i,j]$ at each lattice point:

$$\begin{aligned} \phi [i,j] = \phi ^1 (x(i), y(j)). \end{aligned}$$

(A.4)

While the function $\phi ^1$ lives on the lattice points, its derivatives and the metric are defined to live at the center of the plaquettes. We label the plaquette by the values (i, j) of the lattice point on the lower left of the plaquette. Thus derivatives, for example, are given by

$$\begin{aligned} (\partial _x \phi ) [i,j] =\,&{1\over 2\Delta } \bigl ( \phi [i+1,j] + \phi [i+1, j+1] - \phi [i,j] - \phi [i, j+1]\bigr ), \nonumber \\ (\partial _y \phi ) [i,j] =\,&{1\over 2\Delta } \bigl ( \phi [i,j+1] + \phi [i+1, j+1] - \phi [i,j] - \phi [i+1, j]\bigr ). \end{aligned}$$

(A.5)

Note that on account of (4.12) relating $\phi ^1(=\phi )$ and $\phi ^2$ derivatives we have

$$\begin{aligned} (\partial _x \phi ^2) [i,j] = (\partial _y \phi ) [j,i], \quad \hbox {and} \quad (\partial _y \phi ^2) [i,j] = (\partial _x \phi ) [j,i]. \end{aligned}$$

(A.6)

The value of the scale factor $\Omega$ at the center of the (i, j) plaquette is called $\Omega [i,j]$ and, as indicated in the program (4.18), we have two inequalities to consider:

$$\begin{aligned} \Omega [i,j] \ \ge&\bigl ( 1 + (\partial _x \phi ) [i,j]\bigr )^2 + \bigl ( (\partial _y \phi ) [i,j]\bigr )^2, \nonumber \\ \Omega [i,j] \ \ge&\bigl ( (\partial _y \phi ) [j,i]\bigr )^2 + \bigl ( 1 + (\partial _x \phi ) [j,i]\bigr )^2. \end{aligned}$$

(A.7)

Note that by defining

$$\begin{aligned} S[i,j] \ \equiv \ \bigl ( 1 + (\partial _x \phi ) [i,j]\bigr )^2 + \bigl ( (\partial _y \phi ) [i,j]\bigr )^2, \end{aligned}$$

(A.8)

the inequalities for the metric take the simple form,

$$\begin{aligned} \Omega [i,j] \ge S[i,j], \quad \hbox {and} \quad \Omega [i,j] \ge S[j,i]. \end{aligned}$$

(A.9)

These constraints make it manifest that, as desired, the metric has the expected reflection symmetry $\Omega [i,j] = \Omega [j,i]$. Moreover, $\Omega [i,i] = S[i,i]$. The area functional to be minimized is then given by

$$\begin{aligned} \hbox {Area}= \sum _{i,j\in F} 4 \Delta ^2 \ \Omega [i,j]. \end{aligned}$$

(A.10)

Here F is the set of points labeling plaquettes in the fundamental domain and the factor of four is needed because the Swiss cross contains the fundamental domain four times. More explicitly, the set of values in F can be read from the figure and are:

$$\begin{aligned} F = \{i= 1, \ldots , N, j = 1, \ldots kN\} \cup \{i= N+1, \ldots , kN, j = 1, \ldots N\}. \end{aligned}$$

(A.11)

We need also the Gaussian curvature K of our metric $ds^2 = \Omega (dx^2 + dy^2)$. The value is

$$\begin{aligned} K = - {1\over 2 \Omega } \nabla ^2 \ln \Omega . \end{aligned}$$

(A.12)

In a planar discretization the Laplacian of a function is computed using 5 points, one at the center where we evaluate the Laplacian. There are two natural options, a cross configuration and a square configuration. For a function f(x, y) the Laplacian at the origin using these configurations is

$$\begin{aligned} \nabla ^2 f(0,0) =&{1\over h^2} \Bigl ( f(h,0) + f(-h,0) + f(0,h) + f(0,-h) - 4 f(0,0) \Bigr ), \nonumber \\ \nabla ^2 f(0,0) =&{1\over 2h^2} \Bigl ( f(h,h) + f(-h,-h) + f(h,-h) + f(-h,h) - 4 f(0,0) \Bigr ). \end{aligned}$$

(A.13)

Applied to our lattice $h= 1/(2N)$ and this gives us two options

$$\begin{aligned} K[i,j] =&- {2N^2\over \Omega [i,j]} \ln \biggl ( \ {\Omega [i,j+1] \,\Omega [i, j-1] \, \Omega [i+1, j] \, \Omega [i-1, j] \over \Omega ^4[i,j] }\ \biggr ) , \nonumber \\ K[i,j] =&- {N^2\over \Omega [i,j]} \ln \biggl ( {\Omega [i+1,j+1] \,\Omega [i+1, j-1] \, \Omega [i-1, j+1] \, \Omega [i-1, j-1] \over \Omega ^4[i,j] }\biggr ). \end{aligned}$$

(A.14)

1.2 Convergence

In this “Appendix” we present data showing that the discretized primal and dual programs agree with each other and converge as the lattice resolution is increased. For concreteness, we focus on the $\ell _s=2$ Swiss cross, in the L frame. Very similar convergence behavior was observed for the punctured torus.

Fig. 39

a and b Extremal area A on the $\ell _s=2$ Swiss cross obtained numerically from the primal (blue) and dual (orange) programs, versus the linear resolution “res” of the lattice discretization. c Absolute value of difference between the optimal A at a given resolution and the optimum value $A_*$ in (A.15) (the average of the primal and dual results at the highest resolution). The dotted black line is (res)$^{-3}$, showing that convergence is in excellent agreement with a power law with an exponent around $-3$

We start by studying the convergence of the extremal area as the resolution is increased. We define the linear resolution as the number of plaquettes on the long edge of the fundamental domain. In the notation of subsection A.1 the linear resolution is 2N. The total number of plaquettes in the fundamental domain is $3N^2$.

Figure 39 shows the extremal area computed from the primal and dual programs at linear resolutions ranging from 4 to 128 plaquettes. The highest-resolution value for the extremal area is obtained from either the primal or dual, which give the same result to seven significant figures:

$$\begin{aligned} A_* = 2.806\,970. \end{aligned}$$

(A.15)

The error will be estimated below as $\pm 10^{-6}$, or $\pm 1$ in the last digit.

There are two notable features in the numerical results. The first one is that the optimal values of the primal and the dual track each other quite closely as the resolution changes, with the primal value a tiny bit higher than the dual. At low resolution, both underestimate the extremal area. It might have been expected that the primal program, since it is a minimization program, would overestimate the area. However, while the primal objective evaluated exactly on any continuum configuration bounds the true minimum, in our numerical implementation the configuration is defined on a lattice and the objective involves discretized derivatives. Therefore there is no inconsistency in having a value of the objective which lies below the true continuum minimum. While we do not know why the primal and dual optimal values track each other so closely as the resolution is increased, one plausible explanation is that the discretized programs, at the same resolution, are actually related to each other (perhaps only approximately) by strong duality.

The second notable result is that the primal and dual optimal values both converge to a common value at a rate consistent with a power law with exponent $-3$. Indeed, defining the error $\Delta A$ as a function of the linear resolution “${\mathrm{res}}$” (equal to 2N in the language of “Appendix A.1”)

$$\begin{aligned} \Delta A(\mathrm{res}) := A({\mathrm{res}}) -A_*, \end{aligned}$$

(A.16)

with $A_*$ the highest resolution value in (A.15), we find

$$\begin{aligned} |\Delta A(\mathrm{res})| \approx (\mathrm{{res}})^{-3}. \end{aligned}$$

(A.17)

It would be interesting to try to derive this convergence rate from a first-principles analysis of the discretized programs. We have not attempted such an analysis; instead we merely take the agreement between the primal and dual and their joint rapid convergence as indications that the errors introduced by the discretization are under control and the results are reliable. In this spirit we estimate the error in (A.15) as $\pm 10^{-6}$, calculated by assuming that the error depends on the resolution as in (A.17). A more conservative error estimate is to take the difference between the highest and second-highest resolutions (res $=128$ and res $=64$), about $5\times 10^{-6}$.

Fig. 40

a and b Value of the line element $\rho$ in the extremal metric at a representative point midway between the bottom-left corner and the inner corner of the fundamental domain, in the primal (blue) and dual (orange) programs, versus linear resolution. c Absolute value of difference between $\rho$ and the average of the highest-resolution values. The dotted black line is $\frac{1}{3}$(res)$^{-3}$, showing that convergence is in good agreement with a power law with an exponent around $-3$

Fig. 41

$\ln \rho$ computed from the optimal configuration in the dual program at various linear resolutions: a 4; b 8; c 16; d 32; e 64; f 128

Next, we study the extremal metric, or more precisely its line element $\rho =\sqrt{\Omega }$. In Fig. 40, we show the convergence of $\rho$ for both the primal and dual at a representative point, the midpoint between the bottom-left corner and the inner corner of the fundamental domain (corresponding to $x=y=\tfrac{1}{4}$ in “Appendix A.1”). Just as they did for the area, the metrics obtained from the primal and dual track each other closely and converge as a function of resolution like a power law with an exponent of roughly $-3$. At the highest resolution, the value of $\rho$ at the representative point differs between the primal and dual by less than 1 part in $10^6$. The difference between the primal and dual is not uniform over the fundamental domain, being largest near the inner corner, but is everywhere less than 1 part in $10^3$. Finally, in Fig. 41, we show the convergence of $\ln \rho$ in the dual program over the whole fundamental domain as the resolution is increased.