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Amanda Hirschi

Publications and source records attributed to Amanda Hirschi.

9 recordsLinked to original sources
arXiv2026

Open Gromov-Witten invariants in genus zero and Lagrangian cobordisms

We construct open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians, which are weakly unobstructed by a bounding cochain. This uses the foundational work of \cite{HH25,HH26} and the algebraic framework of \cite{ST21}. We prove the open WDVV relations and show that these invariants are independent of the choice of almost complex structure and under Hamiltonian isotopy. We also prove a relation between open Gromov-Witten invariants of cobordant Lagrangians.

math.SG↗
arXiv2026

Open-closed Deligne-Mumford field theories: construction

Open-closed Deligne--Mumford field theories are chain-level field theories based on moduli spaces of stable curves with boundary. We associate to a relatively spin embedded Lagrangian $L \subset (X,\omega)$ such an open-closed DMFT. It extends the Fukaya $A_\infty$ algebra to curves of arbitrarily high genus and with arbitrarily many boundary components and is unique up to homotopy. This is the first step in proving Kontsevich's conjecture that the Fukaya category determines the Gromov--Witten invariants of $X$, following a strategy delineated by Costello.

math.SG↗
arXiv2025

A contact homotopy type

Adapting the construction of global Kuranishi charts to the contact setting, we associate to any non-degenerate contact manifold a flow category based on Reeb orbits and moduli spaces of pseudo-holomorphic buildings. The construction lifts contact homology and is natural in the sense that to any exact symplectic cobordism we can associate a flow bimodule between the flow categories of its ends.

math.SG↗
arXiv2025

Open-closed Deligne-Mumford field theories: geometric foundations

We construct global Kuranishi charts for moduli spaces of pseudo-holomorphic maps of arbitrary genus with boundary on an embedded Lagrangian submanifold. We then build the geometric foundations required for obtaining compatible chain-level operations, which are employed in follow-up work to construct an open-closed Deligne-Mumford field theory.

math.SG↗
arXiv2024

Lagrangian intersections and cuplength in generalised cohomology theories

We find lower bounds on the number of intersection points between two relatively exact Hamiltonian isotopic Lagrangians. The bounds are given in terms of the cuplength of the Lagrangian in various multiplicative generalised cohomology theories. The intersection of the Lagrangians need not be transverse, however, we require certain orientation assumptions. This gives stronger bounds than previous estimates on the number of self-intersection points of a suitable closed, relatively exact Lagrangian diffeomorphic to Sp$(2)$ or Sp$(3)$. Our proof uses Lusternik-Schnirelmann theory, following and extending work by Hofer.

math.SG↗
arXiv2023

Properties of Gromov-Witten invariants defined via global Kuranishi charts

Using the global Kuranishi charts constructed in \cite{HS22}, we define gravitational descendants and equivariant Gromov-Witten invariants for general symplectic manifolds. We prove that that these invariants, equivariant and non-equivariant, satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it derive an explicit formula for the equivariant GW invariants of a class of Hamiltonian manifolds. A comparison with the GW invariants of \cite{RT97} is given in the semipositive case.

math.SG↗
arXiv2023

On Donaldson's 4-6 question

We prove that the examples by Smith and McMullen-Taubes provide infinitely many counterexamples to one direction of Donaldson's 4-6 question and the closely related Stabilising Conjecture. These are the first known counterexamples. In the other direction, we show that the Gromov-Witten invariants of two simply-connected closed symplectic $4$-manifolds, whose products with $(S^2,\omega_{\text{std}})$ are deformation equivalent, agree. In particular, when $b_2^+ \geq 2$, these $4$-manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace $(S^2,\omega_{\text{std}})$ by $(S^2,\omega_{\text{std}})^k$ for any $k \geq 1$ in both results.

math.SG↗
arXiv2023

Infinitely many monotone Lagrangian tori in higher projective spaces

Vianna constructed infinitely many exotic Lagrangian tori in the complex projective plane. We lift these tori to higher-dimensional projective spaces and show that they remain non-symplectomorphic. Our proof is elementary except for an application of the wall-crossing formula by Pascaleff-Tonkonog.

math.SG↗