Convex Programs for Minimal-Area Problems

Abstract

The closed string field 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 at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem.

Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

“…This type of result is apparently of interest in closed string theory; see the work by Zwiebach and coauthors [42], [43], [17], [18], [27].…”

Minimal-Area Metrics on the Swiss Cross and Punctured Torus

“…This follows from strict convexity of the area functional: if the extremal metric and its transformed version are different, the average metric would have lower area while satisfying all the constraints. The general version of this argument was given in [2], at the end of section 3.1.…”

“…(1.1) The Swiss cross is an excellent problem to try the new programs proposed in [2]. As formulated above, however, it is not a closed string theory minimal-area problem.…”

“…For those values of h, therefore, the extremal metric solves the minimal area problem that also constrains curves homotopic to the boundary. For h < h (2) , however, the minimal area metric with two constrained homologies fails to satisfy the length condition for curves homotopic to the boundary. This condition must now be imposed explicitly.…”

Localization of effective actions in heterotic string field theory

“…However closed string field theory (see [44] for a recent review) is not as explicit as open string field theory because the fundamental vertices defining its interactions necessarily include integrations over implicitlydefined internal regions of the moduli space of punctured Riemann surfaces, together with local coordinates around punctures, for which we generally do not have closed form expressions. But since these data are necessary for constructing off-shell amplitudes, it seems that a direct approach towards analytic computations in closed string field theory is still not really available, although progress in this direction is happening [45][46][47][48][49][50]. However, on second thought, we would expect that when we are concerned with physical quantities, the final result should be independent on the various un-physical data that are needed for the definition of the fundamental vertices.…”