[Paper Review] Invariance of Quantum Rings under Ordinary Flops II: A quantum Leray--Hirsch theorem

Yaun-Pin Lee, Hui-Wen Lin+1 more|arXiv (Cornell University)|Nov 22, 2013

Algebraic structures and combinatorial models8 references5 citations

TL;DR

This paper establishes the invariance of big quantum cohomology rings under ordinary flops of splitting type via a new quantum Leray–Hirsch theorem, which extends the Dubrovin connection on the base to the total space of a toric bundle using a Picard–Fuchs system. The key result is the proof of quantum invariance through analytic continuation with nontrivial Birkhoff factorization, resolving a long-standing problem in crepant transformation conjectures for non-toric local geometries.

ABSTRACT

This is the second of a sequence of papers proving the quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete the proof of the invariance of the big quantum rings under ordinary flops of splitting type. To achieve that, several new ingredients are introduced. One is a quantum Leray--Hirsch theorem for the local model (a certain toric bundle) which extends the quantum D module of Dubrovin connection on the base by a Picard--Fuchs system of the toric fibers. Nonsplit flops as well as further applications of the quantum Leray--Hirsch theorem will be discussed in subsequent papers. In particular, a quantum splitting principle is developed in Part III which reduces the general ordinary flops to the split case solved here.

Motivation & Objective

• To prove the invariance of the big quantum ring under ordinary flops of splitting type, a key case in the crepant transformation conjecture.
• To develop a quantum Leray–Hirsch theorem that lifts the quantum D-module from the base to the total space of a toric bundle.
• To establish analytic continuation with nontrivial Birkhoff factorization in cases where the exceptional locus cannot be deformed to a toric geometry.
• To lay the foundation for a quantum splitting principle in subsequent work, reducing general ordinary flops to the split case.

Proposed method

• Introduce a quantum Leray–Hirsch theorem that extends the quantum D-module of the base via a Picard–Fuchs system on the toric fibers.
• Use Birkhoff factorization to regularize quantum differential equations (QDE) and handle unstable series in the generating functions.
• Apply $π^*$-lifting and hypergeometric modification to construct canonical lifts of the quantum D-module from the base to the total space.
• Utilize $ℚ^*$-localization techniques along fiberwise $π^*$-actions to compute invariants in the split bundle setting.
• Construct a generalized mirror transform that relates the small and big quantum cohomology via a nontrivial regularization of harmonic series.
• Employ harmonic convolution and regularization techniques to cancel divergent terms in the unstable range, ensuring polynomiality of corrections.

Experimental results

Research questions

• RQ1 Can the big quantum cohomology ring be invariant under ordinary flops when the local geometry is not deformable to a toric structure?
• RQ2 How can the quantum D-module of the base be canonically lifted to the total space of a toric bundle in the presence of nontrivial Birkhoff factorization?
• RQ3 What role does the Picard–Fuchs system of the toric fiber play in extending the quantum D-module in the quantum Leray–Hirsch theorem?
• RQ4 How can analytic continuation be achieved with nontrivial Birkhoff factorization in the absence of a classical mirror map?
• RQ5 Can the quantum splitting principle be realized through the construction of a quantum Leray–Hirsch theorem for split flops?

Key findings

• The quantum Leray–Hirsch theorem successfully extends the quantum D-module of the base to the total space of the toric bundle via a Picard–Fuchs system on the fibers.
• The invariance of the big quantum ring under ordinary flops of splitting type is fully established through analytic continuation with nontrivial Birkhoff factorization.
• The correction terms from unstable harmonic series are canceled via harmonic convolution in the difference $\mathscr{F}P_2(z)I^{X/S} - P_2'(z)I^{X'/S}$, ensuring polynomial regularization.
• The generalized mirror transform is constructed to relate the small and big quantum cohomology, with the classical mirror map recovered when $\tau = t$ and $I = J$ on $H^0 \oplus H^2$.
• The proof confirms the first known case of quantum invariance under $K$-equivalence where the exceptional locus is not deformable to a toric geometry and analytic continuation involves nontrivial Birkhoff factorization.
• The method provides a framework for the quantum splitting principle in Part III, reducing general ordinary flops to the split case solved here.