[Paper Review] Derived intersections over the Hochschild cochain complex

Márton Hablicsek|arXiv (Cornell University)|Aug 24, 2016

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper generalizes the BV-algebra structure on derived intersections of Lagrangian submanifolds in the 1-shifted cotangent bundle $T^*[1]X$ of a smooth complex scheme $X$. It shows that the derived intersection of quantized Lagrangians—specifically the zero section and the graph of a closed 1-form—yields the canonical quantization of twisted cotangent bundles, confirming a meta-principle: quantization of derived intersections coincides with derived intersections of quantized Lagrangians.

ABSTRACT

The paper generalizes a result of Behrend-Fantechi and Baranovsky-Ginzburg to the 1-shifted cotangent bundle $T^*X[1]$ of a smooth scheme $X$ over the field of complex numbers. We show how one can obtain twisted cotangent bundles as derived intersections of Lagrangians in $T^*X[1]$, moreover and we show that the derived intersection of the quantized Lagrangians coincide with the canonical quantization of the twisted cotangent bundles.

Motivation & Objective

• To extend the BV-algebra structure on derived intersections of Lagrangians from the standard cotangent bundle to the 1-shifted cotangent bundle $T^*[1]X$.
• To demonstrate that the derived intersection of quantized Lagrangians in $T^*[1]X$ recovers the canonical quantization of twisted cotangent bundles.
• To provide a categorical and homological framework for understanding the duality between derived intersections and quantization in shifted symplectic geometry.
• To verify the meta-principle: the quantization of the derived intersection of Lagrangians equals the derived intersection of the quantized Lagrangians.

Proposed method

• Utilizes the Costello-Li framework for quantization of shifted symplectic structures, particularly the $-1$-shifted and $0$-shifted cases.
• Constructs Lagrangian structures on the zero section $Y = X \to T^*[1]X$ and the graph $Z$ of a closed 1-form $\alpha \in H^1(X, \Omega^1_X)$, requiring $d\alpha = 0$.
• Applies derived algebraic geometry to compute the derived intersection $W = Y \underset{T^*[1]X}{\overset{L}{\times}} Z$, showing it is quasi-isomorphic to a twisted cotangent bundle over $X$.
• Establishes a quasi-isomorphism between the derived tensor product $\mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D_{X}^{\mathrm{op}})^{\mathrm{op}} \otimes_{\mathrm{Diff}({\mathscr{O}}_X^{\bullet}, {\mathscr{O}}_X)} \mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D)$ and the algebra $D$ of differential operators on $X$, using filtered algebra techniques.
• Leverages étale local trivializations and Weyl algebra relations to lift the $D$-algebra structure to the derived tensor product, proving the map is a dg-algebra quasi-isomorphism.
• Uses the trivialization of the canonical bundle $\omega_X \cong {\mathscr{O}}_X$ to relate the derived intersection to twisted de Rham complexes, linking to BV-differentials.

Experimental results

Research questions

• RQ1 How does the BV-algebra structure on derived intersections extend from the standard cotangent bundle to the 1-shifted cotangent bundle $T^*[1]X$?
• RQ2 Can the derived intersection of quantized Lagrangians in $T^*[1]X$ be identified with the canonical quantization of a twisted cotangent bundle?
• RQ3 Under what conditions do the zero section and the graph of a closed 1-form on $T^*[1]X$ admit Lagrangian structures?
• RQ4 Does the derived intersection of these quantized Lagrangians recover the $\hbar$-deformation of the structure sheaf of the intersection?
• RQ5 Is there a general principle that equates the quantization of a derived intersection with the derived intersection of quantized Lagrangians?

Key findings

• The derived intersection $W$ of the zero section and the graph of a closed 1-form $\alpha$ in $T^*[1]X$ is quasi-isomorphic to a twisted cotangent bundle over $X$, with the twist determined by $\alpha$.
• The derived intersection $W$ carries a $0$-shifted symplectic structure, and its quantization is naturally a Batalin-Vilkovisky algebra.
• The derived tensor product $\mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D_{X}^{\mathrm{op}})^{\mathrm{op}} \otimes_{\mathrm{Diff}({\mathscr{O}}_X^{\bullet}, {\mathscr{O}}_X)} \mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D)$ is quasi-isomorphic to the algebra $D$ of differential operators on $X$, as filtered associative algebras.
• The BV-differential on the derived intersection arises from the twisted de Rham complex $0 \to {\mathscr{O}}_X \xrightarrow{d + \wedge d\alpha} \Omega^1_X \to \cdots$, confirming the BV structure.
• The construction confirms the meta-principle: the quantization of the derived intersection of Lagrangians coincides with the derived intersection of the quantized Lagrangians.
• The quasi-isomorphism $\chi: D \to \mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D_{X}^{\mathrm{op}})^{\mathrm{op}} \otimes_{\mathrm{Diff}({\mathscr{O}}_X^{\bullet}, {\mathscr{O}}_X)} \mathrm{Diff}({\mathscr{O}}_X^{\bullet}, D)$ is a dg-algebra map and a filtered quasi-isomorphism, establishing the equivalence at the level of homotopy categories.