Fukaya's Witten-complex conjecture for Lagrangian disc instantons

Let MM be a manifold, let fif_i be functions on MM, and let Li=graph dfiTML_i={\rm graph\ }df_i\subset T^*M for i=1,2,3i=1,2,3. For intersection points pijLiLjp_{ij}\in L_i\cap L_j, let ψij\psi_{ij} be a linear combination of eigenfunctions with sufficiently small eigenvalues of the Witten Laplacian P(k)P^{(k)} on kk-forms associated with fifjf_i-f_j, localized around pijp_{ij}. Fukaya's conjecture. The pairing of these Witten-complex states should satisfy

ψ12ψ23,ψ31L2e1hωψ12ψ23ψ31,\langle \psi_{12}\wedge\psi_{23},\psi_{31}\rangle_{L^2}\sim\sum e^{-\frac{1}{h}\int\omega}\,\lVert\psi_{12}\rVert\,\lVert\psi_{23}\rVert\,\lVert\psi_{31}\rVert,

where the sum is over pseudoholomorphic triangles, or disc instantons, with vertices pijp_{ij}. This is the simplest expected form of an AA_\infty-category structure on Witten complexes associated with different Morse functions, relating their products to Fukaya-type disc instantons.

Sources & referencesView supporting material

Primary source

Alexander Getmanenko, “Resurgent Analysis of the Witten Laplacian in One Dimension”, arXiv:0809.0441 (2010).

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