Mirror symmetry for log Calabi-Yau surfaces I

By Mark Gross, P. Hacking, S. Keel. Publications mathématiques (Bures-sur-Yvette), 2011

290 citations recorded by the source corpus.

We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is constructed as the spectrum of an explicit algebra structure on a vector space with canonical basis and multiplication rule defined in terms of counts of rational curves on Y meeting D in a single point. The elements of the canonical basis are called theta functions. Their construction depends crucially on the Gromov-Witten theory of the pair (Y,D).

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/14b9fba856fdaed2d5b1d8dbf421756d89367b1a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. M. Kontsevich
2. David E. Speyer
3. P. Hacking
4. S. Keel
5. Mark Gross
6. Greg Muller
7. A. Goncharov
8. Li-Chien Shen

In Computational Theory and Mathematics

1. P. Hacking
2. S. Keel
3. M. Kontsevich
4. Mark Gross
5. A. Goncharov
6. David E. Speyer
7. Greg Muller
8. Eric Katz

In Geometry and Topology

1. Mark Gross
2. P. Hacking
3. M. Kontsevich
4. David E. Speyer
5. S. Keel
6. Greg Muller
7. Li-Chien Shen
8. Bernd S Siebert

Citations in the corpus

• Birational geometry of cluster algebras (2013, unknown): “ vectors in a symplectic lattice and the piecewise linear cluster mutation is just like moving worms in the integral affine manifolds central to mirror symmetry for open Calabi-Yau varieties, see e.g., [GHK11]. Before this conversation we had been incorrectly assuming the cluster picture was a very sp”
• The twist for positroid varieties (2016, background): “†The open uniform positroid variety in Gr(k, n) is the complement of a simple normal-crossing canonical divisor, making it an example of an affine log Calabi–Yau variety with maximal boundary, in the sense of [12].”
• Geometry of canonical bases and mirror symmetry (2013, background): “Notice that in our conjectures we give an a priori description of the mirror dual pair of spaces, while in [40] the mirror space is encrypted in the conjecture.”
• Canonical bases for cluster algebras (2014, methodology): “The notion of broken line was introduced in [G09], and developed from the point of view of defining canonical functions in [CPS] and [GHK11].”