Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 4

Special Issue: In Memory of Andrey Todorov, Part 1 of 3

Weak Coupling, Degeneration and Log Calabi-Yau Spaces

Pages: 665 – 738

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n4.a4


Ron Donagi (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Sheldon Katz (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Martijn Wijnholt (Arnold Sommerfeld Center, Ludwig-Maximilians Universität, München, Germany)


We establish a new weak coupling limit in $F$-theory. The new limit may be thought of as the process in which a local model bubbles off from the rest of the Calabi-Yau. The construction comes with a small deformation parameter $t$ such that computations in the local model become exact as $t \to 0$. More generally, we advocate a modular approach where compact Calabi-Yau geometries are obtained by gluing together local pieces (log Calabi-Yau spaces) into a normal crossing variety and smoothing, in analogy with a similar cutting and gluing approach to topological field theories. We further argue for a holographic relation between $F$-theory on a degenerate Calabi-Yau and a dual theory on its boundary, which fits nicely with the gluing construction.


string compactification, $F$-theory, Calabi-Yau, elliptic fibration, semi-stable degeneration, smoothing, variation of Hodge structure

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