Toen Paper 2, Quantization <https://hal.archives-ouvertes.fr/hal-01253020/document>

- Applications:
	- Several structures are induced from common structures on corresponding moduli stacks
	- Quantum groups
		- For $G\in\Alg\Grp$, deform the ring of functions $k[G]$ into noncommutative Hopf algebras $A_{\hbar}$ where $\hbar =0$ recovers $k[G]$. 
		- Relevant stack: $\Bun_G(\pt) \homotopic \BG$, and $A_{\hbar}$ arises as a deformation of $\QCoh(\BG) \cong \Rep(G)$.
		- Quantum groups are deformations of $\QCoh(\Bun_G(\pt))$
	- Skein algebras
		- Take $X = \Hom_\Grp(\pi_1 \Sigma_g, \SL_2)/\Inn\SL_2$ and deform the ring of functions $k[X]$ to $K_{\hbar}(\Sigma)$.
		- Relevant stack: $\Bun_{\SL_2}(\Sigma_g)$
		- Skein algebras are deformations of $\QCoh(\Bun_{\SL_2}(\Sigma_g))$
	- Donaldson-Thomas invariants
		- For $X \in \CY^3$, let  $\mcm_X$ be the moduli space of stable vector bundles (with certain fixed invariants); locally $X \embeds Z$ for some $Z$ as $\crit(f)$ for $f: Z\to \AA^1$; each $f$ defines a sheaf of *vanishing cycles* which glue to a global perverse sheaf $\mce$ of vector spaces which is a *quantization* of $X$: it is a noncommutative deformation of a line bundle of "virtual half-forms".
		- Relevant stack: $\Bun_{\GL_n}(X)$.
		- The perverse sheaf $\mce$ is a deformation of $\QCoh(\Bun_{\GL_n}(X))$.
- Motivation for *higher* stacks:
	- The moduli space $\Vect_n(X)$ is representable by a stack, but the moduli space of *chain complexes* of vector bundles is not.
- History: introduced by Grothendieck.
- Derived deformation theory (80s): moduli spaces can be locally described by Maurer-Cartan elements in a local dg Lie algebra $\lieg_p$.
	- E.g. for a smooth projective variety, $\lieg_p = \cocomplex{C}(X; \TT_X)$  is the complex computing cohomology of the tangent sheaf with a Lie bracket of vector fields.
- The derived part: a consequence is that there exist "virtual" sheaves on moduli spaces
	- Can reconstruct local formal functions for $\mcm$ as $\hat{\OO}_{\mcm, x} \cong H^0(\cocomplex{CE}(\lieg_p)) \cong H^0(\lieg_p; k)$ where $CE(\lieg) \da \hat{\Sym} \qty{ \Sigma\inv\lieg_p\dual}$ is the Chevalley-Eilenberg complex, computed by taking $\cocomplex{\Extalg} \lieg \tensor_k \Ug \covers \Triv_\lieg(k)$ as a projective resolution of the trivial module and applying $\Hom_{\Ug}(\wait, \Triv_\lieg(k))$.
	- The higher cohomology $H^{<0}(\CE (\lieg_p))$ may not vanish, this is the *derived structure*: nontrivial coherent sheaves on a formal neighborhood of $p$.
	These control smoothness and are used to build virtual fundamental classes.
- Alternative definition of derived schemes:
![](attachments/Pasted%20image%2020220221215053.png)
- Derived stacks: quotients of derived schemes by actions of smooth groupoids.
	- Associated to a simplicial object whose levels are derived schemes
	- Example: for $G\actson X$ with $G\in \Alg\Grp$ and $X\in\der\Sch$, take the nerve of the simplicial set $[n] \mapsto X\times G\fiberpower{k}{n}$ to get $[X/G]$.
- $L(X)$: the dg-category of "quasicoherent complexes" on $X\in\der\St$.
	- For $X = \spec A$, $L(A) = \dg\mods{A}$, and more generally $L(X) = \ho\cocolim_{\spec A \to X} L(A)$.
- Canonical object $\LL_X \in L(X)$ which derives $\Omega_{X}$ the sheaf of 1-forms
	- For $X\in \Aff\der\Sch$, $\LL_X = \LL \Omega_{(\wait)}$ is a left derived functor. For general $\der\St$, $\LL_X$ is obtained by gluing $\LL_{X_i}$ at each stage in a simplicial presentation.
- Define a tangent complex as $\TT_X = \LL_X\dual$, and a complex in $\Ch\mods{k}$ of differentials: a sheaf $\cocomplex{\mca}: U \mapsto \globsec{U, \cocomplex{\Extalg}_{\OO_X} \LL_X }$ and a total complex $\mca(X) \da \prod_{i\geq 0} \Sigma^{-i}\mca^i(X)$.
	- Carries a de Rham differential, a Hodge filtration, can define symplectic/Poisson structures