[Paper Review] Quantization and Derived Moduli Spaces I : Shifted Symplectic Structures

Tony Pantev, G. Vezzosi|arXiv (Cornell University)|Nov 14, 2011

Algebraic Geometry and Number Theory29 references9 citations

TL;DR

This paper introduces n-symplectic structures—a derived generalization of symplectic geometry—for derived Artin n-stacks, proving that classifying stacks of reductive groups and derived stacks of perfect complexes carry canonical 2-symplectic structures. It establishes a key result: if X is a Calabi-Yau d-dimensional variety and F is an n-symplectic derived stack, then the derived mapping stack Map(X,F) inherits a canonical (n−d)-symplectic structure, yielding new examples of derived moduli spaces with symplectic structures, including on derived moduli of perfect complexes over Calabi-Yau varieties and manifolds.

ABSTRACT

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-symplectic structures, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see [HAG-II, To2]). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (n − d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on CalabiYau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on CalabiYau surfaces, or simple representations of π1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in [Co, Co-Gw]) on the derived mapping

Motivation & Objective

• To generalize symplectic structures to derived Artin n-stacks via the notion of n-symplectic structures.
• To establish the existence of canonical 2-symplectic structures on classifying stacks of reductive groups and derived stacks of perfect complexes.
• To prove a general existence theorem for (n−d)-symplectic structures on derived mapping stacks Map(X,F) when X is a Calabi-Yau d-fold.
• To recover known smooth moduli spaces with symplectic structures as special cases and extend them canonically to singular loci via 0-symplectic structures.
• To deduce new existence results, such as a natural (−1)-symplectic structure on derived mapping stacks over compact, oriented manifolds.

Proposed method

• Introduce the concept of n-symplectic structures as a derived generalization of symplectic geometry in the context of derived Artin n-stacks.
• Use the framework of higher stacks and derived algebraic geometry, particularly from [HAG-II, To2], to define and analyze n-symplectic structures.
• Apply the Calabi-Yau condition on a variety X of dimension d to induce a shift in the symplectic degree from n to (n−d) on the derived mapping stack Map(X,F).
• Leverage the canonical 2-symplectic structure on the derived stack of perfect complexes and on classifying stacks of reductive groups as foundational examples.
• Construct the derived mapping stack Map(X,F) as a derived moduli space and show that its cotangent complex carries a non-degenerate (n−d)-shifted symplectic form.
• Use formal and geometric arguments to show that previously known symplectic structures on smooth moduli spaces of simple objects extend canonically to singular loci as 0-symplectic structures.

Experimental results

Research questions

• RQ1 How can symplectic structures be generalized to derived algebraic stacks beyond the classical smooth setting?
• RQ2 What conditions on a base variety X ensure that the derived mapping stack Map(X,F) inherits a shifted symplectic structure when F is n-symplectic?
• RQ3 Can known symplectic structures on moduli spaces of simple sheaves or representations be recovered and extended beyond the smooth locus using derived methods?
• RQ4 What are the implications of the Calabi-Yau condition for the shift in symplectic degree on derived mapping stacks?
• RQ5 What new examples of derived moduli stacks with shifted symplectic structures arise from this framework, particularly over Calabi-Yau varieties or manifolds?

Key findings

• The derived stack of perfect complexes and the classifying stack of a reductive group each carry a canonical 2-symplectic structure.
• For any derived Artin stack F with an n-symplectic structure, the derived mapping stack Map(X,F) inherits a canonical (n−d)-symplectic structure when X is a Calabi-Yau d-fold.
• The symplectic structures on smooth moduli spaces of simple sheaves on Calabi-Yau surfaces and simple representations of π₁ of compact Riemann surfaces are recovered as special cases of the general construction.
• These known symplectic structures extend canonically to the entire moduli stack, including singular loci, as 0-symplectic structures.
• A new (−1)-symplectic structure exists on the derived mapping stack over a compact, oriented topological manifold, extending prior formal constructions.
• The framework provides a systematic way to generate new examples of derived moduli stacks with shifted symplectic structures, including on derived moduli of perfect complexes over Calabi-Yau varieties.