Minimal-Area Metrics on the Swiss Cross and Punctured Torus

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. Through every point in such a metric there is a geodesic that saturates the length condition, and 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 minimalarea 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 negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.

Bit threads, Einstein’s equations and bulk locality

“…The maximization above is an example of a convex optimization program and, hence, the equivalence between (1.1) and (1.2) can also be proved using techniques borrowed from convex optimization [31]. Soon after this paper appeared, it was realized that various geometric problems could likewise be translated to the realm of convex optimization leading to interesting new results [32,33]. The connection with convex optimization has also helped uncover various properties of entanglement entropy from the bit thread perspective [34], as well as some generalizations and applications to other entanglement related quantities [35][36][37][38][39][40][41].…”

Convex Programs for Minimal-Area Problems

“…We have not been able to 'solve' the constraint (7.29) to find a parameterization of the allowed metrics in a two-band region. Our results in [22] indicate that both positive and negative Gaussian curvature are possible.…”

“…The constraint (7.20) now translates into 22) meaning that at any point on the surface the sum of geodesic densities over all the classes that go through that point is the same.…”

“…Here we will explain how to construct a double cover with this property and thereby constrain the lengths of such curves by calibrations. By a straightforward generalization, it is also possible to construct a higher-multiplicity cover, and in certain cases this may be convenient, for example to preserve symmetries of the surface; we will see an example in [22].…”

Open-closed hyperbolic string vertices

“…The requirement is called geometric master equation and the job of finding the solutions is of fundamental interest in the framework of string field theory. In the past, such solutions were found using various metrics, an example being minimal area metrics [12][13][14][15][16]. There were also approximate constructions using the hyperbolic metrics [17][18][19][20].…”