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: 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 \(G\curvearrowright X\) 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