# 003 Toen Paper 2 Quantization

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 $$G\in{\mathsf{Alg}}{\mathsf{Grp}}$$, deform the ring of functions $$k[G]$$ into noncommutative Hopf algebras $$A_{\hbar}$$ where $$\hbar =0$$ recovers $$k[G]$$.
• Relevant stack: $${\mathsf{Bun}}_G({\operatorname{pt}}) \simeq{{\mathbf{B}}G}$$, and $$A_{\hbar}$$ arises as a deformation of $${\mathsf{QCoh}}({{\mathbf{B}}G}) \cong {\mathsf{Rep}}(G)$$.
• Quantum groups are deformations of $${\mathsf{QCoh}}({\mathsf{Bun}}_G({\operatorname{pt}}))$$
• Skein algebras
• Take $$X = \mathop{\mathrm{Hom}}_{\mathsf{Grp}}(\pi_1 \Sigma_g, {\operatorname{SL}}_2)/\mathop{\mathrm{Inn}}{\operatorname{SL}}_2$$ and deform the ring of functions $$k[X]$$ to $$K_{\hbar}(\Sigma)$$.
• Relevant stack: $${\mathsf{Bun}}_{{\operatorname{SL}}_2}(\Sigma_g)$$
• Skein algebras are deformations of $${\mathsf{QCoh}}({\mathsf{Bun}}_{{\operatorname{SL}}_2}(\Sigma_g))$$
• Donaldson-Thomas invariants
• For $$X \in { \text{CY} }^3$$, let $${\mathcal{M}}_X$$ be the moduli space of stable vector bundles (with certain fixed invariants); locally $$X \hookrightarrow Z$$ for some $$Z$$ as $$\operatorname{crit}(f)$$ for $$f: Z\to {\mathbb{A}}^1$$; each $$f$$ defines a sheaf of vanishing cycles which glue to a global perverse sheaf $${\mathcal{E}}$$ of vector spaces which is a quantization of $$X$$: it is a noncommutative deformation of a line bundle of “virtual half-forms”.
• Relevant stack: $${\mathsf{Bun}}_{\operatorname{GL}_n}(X)$$.
• The perverse sheaf $${\mathcal{E}}$$ is a deformation of $${\mathsf{QCoh}}({\mathsf{Bun}}_{\operatorname{GL}_n}(X))$$.
• Motivation for higher stacks:
• The moduli space $${ \mathsf{Vect} }_n(X)$$ 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 $${\mathfrak{g}}_p$$.
• E.g. for a smooth projective variety, $${\mathfrak{g}}_p = { {C}^{\scriptscriptstyle \bullet}} (X; {\mathbb{T}}_X)$$ 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 $${\mathcal{M}}$$ as $$\widehat{{\mathcal{O}}}_{{\mathcal{M}}, x} \cong H^0( { {CE}^{\scriptscriptstyle \bullet}} ({\mathfrak{g}}_p)) \cong H^0({\mathfrak{g}}_p; k)$$ where $$CE({\mathfrak{g}}) \coloneqq\widehat{\operatorname{Sym}} \qty{ \Sigma^{-1}{\mathfrak{g}}_p {}^{ \vee }}$$ is the Chevalley-Eilenberg complex, computed by taking $${ { { {\bigwedge}^{\scriptscriptstyle \bullet}} }^{\scriptscriptstyle \bullet}} {\mathfrak{g}}\otimes_k {\mathcal{U}(\mathfrak{g}) }\rightrightarrows\mathop{\mathrm{Triv}}_{\mathfrak{g}}(k)$$ as a projective resolution of the trivial module and applying $$\mathop{\mathrm{Hom}}_{{\mathcal{U}(\mathfrak{g}) }}({-}, \mathop{\mathrm{Triv}}_{\mathfrak{g}}(k))$$.
• The higher cohomology $$H^{<0}({\mathsf{CE}}({\mathfrak{g}}_p))$$ 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:
• Derived stacks: quotients of derived schemes by actions of smooth groupoids.
• Associated to a simplicial object whose levels are derived schemes
• Example: for $$G\curvearrowright X$$ with $$G\in {\mathsf{Alg}}{\mathsf{Grp}}$$ and $$X\in{\mathsf{d}}{\mathsf{Sch}}$$, take the nerve of the simplicial set $$[n] \mapsto X\times G{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {k} }{\times} }^{n} } }$$ to get $$[X/G]$$.
• $$L(X)$$: the dg-category of “quasicoherent complexes” on $$X\in{\mathsf{d}}{\mathsf{St}}$$.
• For $$X = \operatorname{Spec}A$$, $$L(A) = {\mathsf{dg}}{\mathsf{A}{\hbox{-}}\mathsf{Mod}}$$, and more generally $$L(X) = {\mathsf{ho}}\cocolim_{\operatorname{Spec}A \to X} L(A)$$.
• Canonical object $${\mathbb{L}}_X \in L(X)$$ which derives $$\Omega_{X}$$ the sheaf of 1-forms
• For $$X\in {\mathsf{Aff}}{\mathsf{d}}{\mathsf{Sch}}$$, $${\mathbb{L}}_X = {\mathbb{L}}\Omega_{({-})}$$ is a left derived functor. For general $${\mathsf{d}}{\mathsf{St}}$$, $${\mathbb{L}}_X$$ is obtained by gluing $${\mathbb{L}}_{X_i}$$ at each stage in a simplicial presentation.
• Define a tangent complex as $${\mathbb{T}}_X = {\mathbb{L}}_X {}^{ \vee }$$, and a complex in $$\mathsf{Ch}{\mathsf{k}{\hbox{-}}\mathsf{Mod}}$$ of differentials: a sheaf $${ {{\mathcal{A}}}^{\scriptscriptstyle \bullet}} : U \mapsto {{\Gamma}\qty{U, { { { {\bigwedge}^{\scriptscriptstyle \bullet}} }^{\scriptscriptstyle \bullet}} _{{\mathcal{O}}_X} {\mathbb{L}}_X } }$$ and a total complex $${\mathcal{A}}(X) \coloneqq\prod_{i\geq 0} \Sigma^{-i}{\mathcal{A}}^i(X)$$.
• Carries a de Rham differential, a Hodge filtration, can define symplectic/Poisson structures