Chapter 3: manifestly rational universal quantization
3.4
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A cosheaf on R that “smears out” the universal formal Poisson manifold
In this section Q we will describe what we mean by “smearing out” the universal formal Poisson manifold A = (X ⊗n /Sn ) across R. We will end up assigning to each open set U ⊆ R a proobject A(U ) ∈ DGVect[X, π]. If U is an interval, then A(U ) will be homotopic to A. The precosheaf A(−) will satisfy the conditions of a prefactorization algebra (Definition 3.4.5). At the end of the section we will discuss the problem of deforming or “quantizing” this prefactorization algebra. 3.4.1 Definition (A(−)) For each open U ⊆ R and for each n, in the previous section we defined a chain complex C• (U n ) ∈ DGVect, with an action of Sn permuting the n copies of U . Functoriality for diagonal maps implies that U 7→ C• (U n ) is a precosheaf of Sn -modules. We consider then the object C• (U n ) ⊗ X ⊗n ∈ DGVect[X, π], and give it the diagonal Sn action, and define: Y A(U ) = C• (U n ) ⊗ X ⊗n /Sn n≥0
Experts may recognize this as a particular model for factorization homology with coefficients in A. Combining the graphical notations of the previous sections, we set U = U = C• (U )⊗X, and · · · =
···
= C• (U n ) ⊗ X ⊗n . As in 3.2.3, denote the map that averages for the
n
Sn action by a horizontal bar, and identify objects with the projections that pick them out. Then: n ...
A(U ) =
Y ...
n≥0 UU
U
3.4.2 Remark (A(−) is valued in commutative algebras) Recall that C• comes equipped with canonical maps ⊗ : C• (U m ) ⊗ C• (U n ) ,→ C• (U m+n ), compatible with the symmetric group actions. We define
: (C• (U m ) ⊗ X ⊗m )/Sm ⊗ (C• (U n ) ⊗ X ⊗n )/Sn → (C• (U m+n ) ⊗ X ⊗m+n )/Sm+n by = avem+n ◦⊗. Summing the various maps gives a map A(U ) ⊗ A(U ) → A(U ). In this way we make A(U ) into a (pro, dg) commutative algebra, and A(−) into a precosheaf of commutative algebras. Before describing further structure, it is worth pointing out the behavior of A(−) on disjoint unions. Let U and V be disjoint open sets in R. Then C• (U m ) ⊗ C• (V m ) ,→