[Paper Review] Algebraic virtual cycles for quantum singularity theories

Huai-Liang Chang, Young Hoon Kiem+1 more|arXiv (Cornell University)|Jun 1, 2018

Algebraic Geometry and Number Theory9 references6 citations

TL;DR

This paper constructs algebraic virtual cycles for quantum singularity theories using a Fourier-Mukai type integral transformation, enabling the cohomological field theory of Fan-Jarvis-Ruan-Witten (FJRW) invariants to be realized via algebraic cycles. By resolving singularities through blowups and applying cosection localization, the authors establish a virtual cycle on a proper moduli space that computes FJRW invariants in both narrow and broad sectors.

ABSTRACT

We construct algebraic virtual cycles that give us the cohomological field theories of Fan-Jarvis-Ruan invariants by integral transformations.

Motivation & Objective

• To provide a purely algebraic construction of cohomological field theories for quantum singularity theories using algebraic cycles.
• To resolve the issue of non-properness in the moduli space of G-spin curves by introducing a blowup of the base space at the singular point.
• To extend the cosection localization principle to singular base spaces by lifting data to a smooth blowup, enabling virtual cycle construction.
• To realize FJRW invariants via an integral transformation using an algebraic virtual cycle, thus bridging matrix factorization theories with classical algebraic geometry.
• To generalize previous results on narrow sectors to broad sectors by constructing a virtual cycle in intersection homology via resolution of singularities.

Proposed method

• Blow up the singular base space $ Z $ at the origin to obtain a smooth $ Z' $, lifting all geometric data to $ Y' $, $ E' $, $ s' $, $ ho' $, and $ ilde{p} $.
• Construct a smooth morphism $ oldsymbol{q}': Y' o Z' $ and define $ X' $ as the zero locus of $ s' $, with $ S' = X' imes_{Z'} S $ proper.
• Apply the cosection localization principle from [12] to the smooth setting, yielding a localized virtual cycle $ [X']^{\mathrm{vir}}_{\mathrm{loc}} \in A_*(S') $.
• Use the virtual pullback $ s^{!}_{\sigma} $ and the refined Gysin map to define the pushforward of the virtual cycle to $ S $, ensuring compatibility with the original invariants.
• Establish a commutative diagram involving $ H_*(Z') \to H_*(S') \to H_*(S) $, showing that the FJRW invariants arise as pushforwards via the integral transformation.
• Prove that the virtual cycle $ [X']^{\mathrm{vir}}_{\mathrm{loc}} $ satisfies the required compatibility with the original invariants through the projection formula and fiber diagram identities.

Experimental results

Research questions

• RQ1 Can the cohomological field theory of FJRW invariants be constructed via an algebraic virtual cycle in both narrow and broad sectors?
• RQ2 How can the cosection localization principle be extended to singular base spaces in the context of G-spin curves?
• RQ3 What role does the blowup of the base space $ Z $ at the origin play in enabling a proper virtual cycle construction?
• RQ4 Is it possible to realize FJRW invariants as the pushforward of a virtual cycle under an integral transformation in algebraic geometry?
• RQ5 How does the virtual cycle on the resolved space $ X' $ relate to the original non-proper moduli space $ X $ via proper pushforward?

Key findings

• The authors construct a virtual cycle $ [X']^{\mathrm{vir}}_{\mathrm{loc}} \in A_*(S') $ on the proper moduli space $ S' $ via cosection localization after blowing up the base space $ Z $ at the origin.
• The virtual cycle $ [X']^{\mathrm{vir}}_{\mathrm{loc}} $ is given by the formula $ s^{!}_{\sigma} [Y'] = [Y'] \cap e_{\sigma}(E', s') $, which is compatible with the Gysin map and the blowup structure.
• The pushforward $ p_*([X']^{\mathrm{vir}}_{\mathrm{loc}}) $ equals the virtual cycle $ [X]^{\mathrm{vir}}_{\mathrm{loc}} $ on $ S $, ensuring the invariants are preserved under proper morphism.
• The FJRW invariants are realized as the pushforward of the virtual cycle via an integral transformation, with the commutative diagram in Theorem 5.4 confirming consistency with the original theory.
• The construction extends previous results from narrow sectors to broad sectors by resolving singularities and applying intersection homology techniques.
• The key identity $ p_*([X']^{\mathrm{vir}}_{\mathrm{loc}} \cap q^*\alpha) = s^{!}_{\sigma} \mathbf{q}^* v $ confirms that the integral transformation computes the correct invariants in $ H_*(S) $.