2023-05-07 Sun

• SBim, character sheaves, mixed geometry
• some Weinstein and Contact geometry from this HDHF paper.

• paper by Ben-Zvi and Nadler: https://arxiv.org/pdf/0904.1247.pdf
• paper by Ho and Li on mixed geometry: https://arxiv.org/pdf/2202.04833.pdf
• paper by Ho-Li on HOMFLY-PT and Hilb on $\C^2$: https://arxiv.org/pdf/2305.01306.pdf
• paper by Gorsky-Hogencamp-Wedrich, https://arxiv.org/pdf/2002.06110.pdf
• paper by Oblomkov-Rozansky, about matrix factorization, B-model realization of the Hecke category.

• we choose orientation for each strands in each Lagrangians.
• I don't know what is a relative spin structure.
• something about capping for each intersection.

Their Lagrangians are union of spheres. They want a trivial $\R$ bundle on the space, so that the direct sum of $\R$ with the tangent bundle is trivial. (well, sometimes, a bundle is stably trivial, you just need to give it more rooms).

We choose trivialization $t$ and $t'$ for the two 'augmented' Lagrangian (are they jet bundle?).

A capping Lagrangian path in the oriented Lagrangians Grassmannian. (so, do I choose orientation of my Lagrangians? I guess I did, since I have trivializations, and I can substract that trivial line bundle.)

A stable capping trivialization is a trivialization of the capping path $L_{p,t} \oplus \R$, that interpolate the original trivializations on the two strands.

OK, these are the data. How do I provides orientation for the moduli spaces then?

We map the upper half-plane to the target space. actually, it is a constant map (what?) to $p$, and we consider the trivial vector bundle $\pi_p^*(T_p M)$. We define the Cauchy-Riemann tuple. $\xi, \eta, D$

• $\xi$ is the symplectic constant vector bundle over $H$.
• on the boundary $\d H$, we segment it into $(-\infty, 0), (0,1), (1,\infty)$, and we put the oriented Lagrangian sub-bundle $\eta$ over $\d H$.
• A Cauchy-Riemann operator, that takes as input a section in the symplectic bundle $\xi$, and output its dbar derivative? OK, I am not sure.

well, a determinant line is a kernel minus cokernel, taking determinants.

If $L$ satisfies certain topological condition, then the moduli space of disks ending on $L$ is orientable.

But, how do we define the A-infinity structures? Why we count some disks with a negative sign? When we say we orient the 0-dim moduli space of the disks, that means plus-minus signs.