Peng Zhou

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====== 2025-02-26 ======

The 3d GMW theory. 

In Kapranov-Kontsevich-Soibelmann paper, 10 years ago, they mentioned that it is possible to consider marked polytope in $\R^3$. There is a $E_3$-algebra controlling the deformation of $E_2$-algebra, and there can be coefficient enhancing all these. I want to understand what precisely is the statement. 


===== 2d GMW =====
To a vertex, we assign a spherical $E_1$-algebra, That means, to each point of $S^1$, we have a algebra (endomorphism of thimbles in that direction). it is locally constant, but can have mutation jumps. 

To the line segment $[p,q]$ from $p$ to $q$, we assign $Hom(p,q)$, which is a $(A_q, A_p)$-bimodule. 
We assume some duality of $Hom(p,q)$ and $Hom(q,p)$. 

To each convex subpolytope, we assign some element in the cyclic tensor product of the bimodules, the boundary of rigid instantons. 

===== 3d GMW =====
To each $0$-simplex, p,  we assign a spherical $E_2$-algebra, $N_p$. 

To each $1$-simplex, $[p,q]$, connecting two points, we assign an $(N_p, N_q)$-bimodule, that is also a spherical $E_1$-algebra. 

To each $2$-simplex, $[p,q,r]$, we may first form the boundary algebra, which is $[p,q] \otimes [q,r] \otimes [r,p] \otimes cyc$. the algebra in the normal direction. We assign an module for this boundary algebra. This time it is just a module (there is no $E_0$-alg)

Then, to each convex marked subpolytope, we need to tensor the boundary modules together, and get a complex, and 'rigid 3-ball' deposit stuff in it.


But, forgetting the coefficient, what do we get from this configuration of points in $\R^3$? 

Well, consider all possible ways to project them to $\R^2$ at $\infty$, that is $S^2$ many ways. For each such half-space (generic). Each point in $A$ generate a ray. pick a finite subset of $A$, we can take convex hulls of the rays. These convex hull are generators of the boundary $E_2$-algebra.