Lagrangian submanifolds from tropical hypersurfaces

Matessi, Diego

doi:10.1142/s0129167x21500464

Int. J. Math.

2021

DOI: 10.1142/s0129167x21500464

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Lagrangian submanifolds from tropical hypersurfaces

Diego Matessi

Abstract: We prove that a smooth tropical hypersurface in [Formula: see text] can be lifted to a smooth embedded Lagrangian submanifold in [Formula: see text]. The idea of the proof is to use Lagrangian pairs of pants, which are the lifts of tropical hyperplanes introduced by the author in an earlier paper, as the main building blocks.

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Cited by 2 publications

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Tropical Quantum Field Theory, Mirror Polyvector Fields, and Multiplicities of Tropical Curves

Mandel

Ruddat

2021

International Mathematics Research Notices

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We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov–Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably these structures include a tropical quantum field theory and an $L_{\infty }$-structure. The latter is an instance of Getzler’s gravity algebra, and the $l_2$-bracket is a restriction of the Schouten–Nijenhuis bracket. We explain the relationship to string topology in the Appendix (thanks to Janko Latschev).

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Tropical Quantum Field Theory, Mirror Polyvector Fields, and Multiplicities of Tropical Curves

Mandel

Ruddat

2021

International Mathematics Research Notices

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Lagrangian Pairs of Pants

Matessi

2019

International Mathematics Research Notices

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We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of an exact one form on the real blowup of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $( {\mathbb{C}}^*)^n$ that lift tropical subvarieties in $\mathbb R^n$. As an example we explain how to lift tropical curves in $ {\mathbb{R}}^2$ to Lagrangian submanifolds of $( {\mathbb{C}}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.

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Lagrangian submanifolds from tropical hypersurfaces

Cited by 2 publications

References 22 publications

Tropical Quantum Field Theory, Mirror Polyvector Fields, and Multiplicities of Tropical Curves

Tropical Quantum Field Theory, Mirror Polyvector Fields, and Multiplicities of Tropical Curves

Lagrangian Pairs of Pants

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