String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

Abstract

This paper introduces new operations on the string topology of a smooth manifold: gravitational descendants of its cotangent bundle, which are augmentations of the Chas-Sullivan $L_{\infty }$ algebra structure of the loop space. The definition extends to Liouville domains. Descendants of the $n$-torus are computed. To a monotone Lagrangian torus in a symplectic manifold, one associates a Laurent polynomial called the Landau-Ginzburg potential, by counting holomorphic disks. This paper proves the following mirror symmetry prediction: the constant terms of the powers of an LG potential are equal to descendant Gromov-Witten invariants of the ambient manifold.

46 pages, 6 figures; v3: important mistake fix (the tangency and descendant GW invariants do not agree), main result is unchanged

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• Functors and Computations in Floer homology with Applications Part II
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Minkowski Polynomials and Mutations
• Duality between Lagrangian and Legendrian invariants
• Higher string topology operations
• Landau-Ginzburg models -- old and new
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• Higher symplectic capacities
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Cited by in corpus (7)

• From symplectic cohomology to Lagrangian enumerative geometry
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Higher symplectic capacities
• Counting curves with local tangency constraints
• On the embedding complexity of Liouville manifolds
• Fano mirror periods from the Frobenius structure conjecture
• Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period