Quantum flips after Lee–Lin–Wang: globalizing the quantum-cohomology embedding under simple flips, and the accompanying singularity, uniqueness, linearity, and Cayley-combinatorics questions

Lee, Lin and Wang (arXiv:1912.03012; Proc. Sympos. Pure Math. 103.2, 2021) compared the big quantum cohomology of the two local models of a simple flip and constructed, for the local models, an $F$ -embedding of the big quantum cohomology of one side into that of the other, with an eigenbundle decomposition of the Dubrovin connection. Their paper leaves open a coherent set of questions that jointly target functoriality of quantum cohomology under K-inequivalent (flip) birational maps: extending the construction from local models to general global simple flips; handling or characterizing the essential singularities of the analytically continued quantum connection; uniqueness of the embedding without an auxiliary normalization; linearity of the embedding in the extremal-ray variable for general $(r,r')$ flips; and a direct localization/graph-sum explanation of the appearance of Cayley numbers $d^{d-2}$ .

Background

Gromov–Witten (GW) invariants are rational counts of stable maps from curves into a fixed smooth projective variety $X$ ; their genus-zero three-point structure constants define the quantum product $\ast$ on the cohomology $H^{\ast}(X)$ , deforming the usual cup product by Novikov variables $q$ that record curve classes. This makes $H^{\ast}(X)$ into the small quantum cohomology ring; allowing the product to depend on the full coordinate vector on $H^{\ast}(X)$ gives big quantum cohomology, governed by the Dubrovin (quantum) connection $\nabla$ , a flat meromorphic connection on the trivial bundle with fiber $H^{\ast}(X)$ over the base of quantum parameters. A central structural question is which of these data are invariant, or transform functorially, under birational maps. For flops — birational maps that are isomorphic in codimension one and $K$ -equivalent, i.e. they identify the canonical divisors — a series of works culminating around 2010 established that genus-zero GW theory is preserved up to a coordinate change, making quantum cohomology a birational invariant within a $K$ -equivalence class. Flips are the $K$ -inequivalent counterparts: contractions of an extremal ray $\ell$ (an extremal ray of the Mori cone of effective curves) on which the canonical class is negative, and they are the elementary steps of the minimal model program that have no flops analogue. Because the canonical class jumps across a flip, quantum cohomology cannot be invariant; the meaningful question is whether it transforms by a computable functorial rule. Lee–Lin–Wang studied the simplest class, the simple $(r,r')$ flips: birational maps $X\dashrightarrow X'$ whose exceptional centers are $Z\cong\mathbf{P}^{r}\subset X$ and $Z'\cong\mathbf{P}^{r'}\subset X'$ with normal bundles $N_{Z/X}\cong\mathcal{O}_{\mathbf{P}^{r}}(-1)^{r'+1}$ and $N_{Z'/X'}\cong\mathcal{O}_{\mathbf{P}^{r'}}(-1)^{r+1}$ , realized by blowing up $X$ along $Z$ and contracting the exceptional divisor $E\cong\mathbf{P}^{r}\times\mathbf{P}^{r'}$ in the other direction. Their projective local models are $X=\mathbf{P}_{\mathbf{P}^{r}}(\mathcal{O}(-1)^{r'+1}\oplus\mathcal{O})$ and $X'=\mathbf{P}_{\mathbf{P}^{r'}}(\mathcal{O}(-1)^{r+1}\oplus\mathcal{O})$ , with $r>r'>0$ , $d:=r-r'$ , both of total dimension $r+r'+1$ ; the extremal curve class $\ell$ (a line in $Z$ ) satisfies $K_X\cdot\ell=-(r-r')<0$ . Their main local theorem constructs a formal $F$ -embedding $\widehat{\Psi}$ — a non-linear embedding of the big quantum cohomology of $X'$ into that of $X$ in the category of $F$ -manifolds — together with a change of variables $\sigma_0(q')$ , and an eigenbundle decomposition of the composite flat connection $\Phi\nabla$ into a regular eigenbundle of rank equal to $\operatorname{rank}H^{\ast}(X')$ and its orthogonal complement, an irregular eigenbundle of rank $d$ ; the construction passes through a Birkhoff factorization (BF, the extraction of a flat section expansion from a fundamental matrix of the connection) and a generalized mirror transform (GMT, the change between the flat coordinates and the naive quantum coordinates, computed by the BF of a hypergeometric $I$ -function). Two analytic phenomena forced the authors into workarounds that generate the open questions. First, $\Phi\nabla$ has essential singularities along $q^{\ell'}=0$ (the divisor where the Novikov variable of the extremal ray $\ell'$ of $X'$ vanishes) in the naive coordinates, so the theory is developed for small quantum cohomology over a cyclic extension of the Novikov ring $R'$ rather than for the big theory; the authors note (their footnote 1) that an adic topology modeled on the Mori cone near $q^{\ell'}=0$ is a plausible systematic substitute, but they do not pursue it. Second, uniqueness of $\sigma_0(q')$ and hence of $\widehat{\Psi}$ is proven only under an auxiliary normalization (imposing their equation (0.3) at $s=0$ ); they expect uniqueness to hold without it. Combinatorially, in the worked $(2,1)$ example (a fourfold, $r'=1$ ), they prove (Theorem 6.12) that for every curve degree $d\ge 1$ the extremal GW invariant with $d+1$ insertions of the class $\kappa_0\in H^{2(r'+1)}(X)=H^{4}(X)$ of the exceptional center $Z$ , in curve class $d\ell$ , equals $\langle\kappa_0^{\otimes(d+1)}\rangle^{X}_{d\ell}=d^{d-2}$ — the Cayley number counting spanning trees of the complete graph $K_d$ on $d$ vertices; here $d$ is the free curve degree, not the kernel rank $r-r'$ (which equals $1$ in this example) — and ask (Remark 6.13) whether a direct equivariant-localization (graph-sum) proof exhibits this tree-counting combinatorially. All statements are proved for the local models and worked explicitly for the $(2,1)$ flip; the paper states that a large part of the arguments extends to more general flips with explicit local structure (e.g. toric flips), but no global theorem is produced. The context of the broader program, including the contrast with the $K$ -equivalence (flop) conjectures, is surveyed in the authors' earlier overview arXiv:1705.04799.

Problem Statement

The common objective is functoriality of genus-zero quantum cohomology under simple flips: turn the Lee–Lin–Wang local-model theory into a computable transformation law attached to a general flip, and clarify the analytic and combinatorial structures it depends on. Within that objective, the source (arXiv:1912.03012) poses the following linked questions. (1) Globalization: construct, for general global simple flips (not only the projective local models $X=\mathbf{P}_{\mathbf{P}^{r}}(\mathcal{O}(-1)^{r'+1}\oplus\mathcal{O})$ and $X'=\mathbf{P}_{\mathbf{P}^{r'}}(\mathcal{O}(-1)^{r+1}\oplus\mathcal{O})$ , $r>r'>0$ ), an analogue of the $F$ -embedding $\widehat{\Psi}$ of the big quantum cohomology of $X'$ into that of $X$ , with the accompanying eigenbundle decomposition of the composite connection $\Phi\nabla$ , identifying the precise analytic hypotheses and mirror (GMT) data needed; the authors note that a large part of the local arguments should extend to flips with explicit local structure such as toric flips, but no global theorem is given. (2) Analytic framework: give a systematic treatment of the essential singularities of $\Phi\nabla$ along $q^{\ell'}=0$ that removes the need for the smallness and cyclic-extension workarounds, or characterize when the essential singularities can be removed, so that the big quantum cohomology of the flip sides can be compared directly. (3) Uniqueness: prove (or disprove) that the change of variables $\sigma_0(q')$ and the $F$ -embedding $\widehat{\Psi}$ are unique without the auxiliary normalization given by imposing equation (0.3) at $s=0$ , as the authors expect (footnote 2). (4) Linearity: for the $(2,1)$ flip the embedding is linear in the extremal-ray variable though necessarily non-linear in the full parameter space; determine whether the embedding is always linear in the extremal-ray variable for general $(r,r')$ flips (Remark 6.14). (5) Cayley combinatorics: for the $(2,1)$ local model, find a direct equivariant-localization (graph-sum) proof that the extremal invariants $\langle\kappa_0^{\otimes(d+1)}\rangle^{X}_{d\ell}$ — with $d+1$ insertions of the class $\kappa_0$ of the exceptional center $Z$ , in curve class $d\ell$ — equal the Cayley number $d^{d-2}$ for every curve degree $d\ge 1$ (Theorem 6.12; here $d$ is the curve degree, not $r-r'$ ), exhibiting the spanning-tree combinatorics of the complete graph $K_d$ geometrically (Remark 6.13). Items (1)-(5) are independent questions rather than satellites of item (1): resolving any one of them, including item (1) on its own, is a partial result, and a complete answer to the shared objective requires all five.

References

Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, Quantum flips I: local model, arXiv:1912.03012, https://arxiv.org/abs/1912.03012; published as Proc. Sympos. Pure Math. 103.2, AMS, 2021, pp. 303-352, DOI 10.1090/pspum/103.2/01861, https://doi.org/10.1090/pspum/103.2/01861
Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, Quantum cohomology under birational maps and transitions (survey of the program), arXiv:1705.04799, https://arxiv.org/abs/1705.04799
Hiroshi Iritani, Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 16 (2020), article 032, DOI 10.3842/SIGMA.2020.032, https://doi.org/10.3842/SIGMA.2020.032
Yefeng Shen, Mark Shoemaker, Quantum spectrum and Gamma structure for standard flips, arXiv:2502.08762, https://arxiv.org/abs/2502.08762
Zhaoxing Gu, Song Yu, Tony Yue Yu, Quantum cohomology of variations of GIT quotients and flips, arXiv:2508.15770, https://arxiv.org/abs/2508.15770