Minimal-Area Metrics on the Swiss Cross and Punctured Torus
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220_2020_3734_ReferencePDF.pdf
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15.02 MB
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Unknown
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40d7d3e84171a5dc20f8ce6ef7725a37
Author(s)
Headrick, Matthew
Zwiebach, Barton
Date Issued
March 2020
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00220-020-03734-z
