The holonomic-module model conjecture for the Fukaya category

Let XX be a symplectic manifold, let Fnaive(X,ω)F^{naive}(X,\omega) denote its naive Fukaya category, and let Db(hol(X))D^b_{\infty}(hol(X)) be the derived AA_{\infty}-category of holonomic modules over the quantized algebra of smooth functions on XX. For Lagrangian submanifolds LiL_i and holonomic modules Mi=V(Li,ρi)M_i=V_{(L_i,\rho_i)}, suppose that the kernels K(Li,Lj)K(L_i,L_j) and the higher compositions mnnewm_n^{new} are chosen as described in the preceding construction. Holonomic-module model conjecture. (i) The higher compositions mnnewm_n^{new}, for n1n\geq 1, give rise to a structure of an AA_{\infty}-category on Db(hol(X))D^b_{\infty}(hol(X)). (ii) This category is AA_{\infty}-equivalent to Fnaive(X,ω)F^{naive}(X,\omega), with q=tq=t. This conjecture proposes an algebraic replacement of instanton counting in the Fukaya category by homomorphisms between holonomic modules and local systems supported on Lagrangian submanifolds; the source does not state whether it has been resolved.

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Primary source

Paul Bressler and Yan Soibelman, “Mirror symmetry and deformation quantization”, arXiv:hep-th/0202128 (2002).

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