Regularized quantum period
- Quantum period and regularization
- The ordinary quantum period $\mathrm{G}_X$ encodes genus-zero, one-point descendant Gromov–Witten invariants.
The hat denotes factorial regularization.
$$\begin{equation} \notag \begin{gathered} \mathrm{G}_X(t)=\sum_{d\geq0}a_dt^d, \qquad \widehat{\mathrm{G}}_X(t)=\sum_{d\geq0}c_dt^d, \qquad c_d=d!\,a_d, \\ a_0=1,\quad a_1=0, \qquad a_d=\sum_{\substack{\beta\in \mathrm{H}_2(X,\mathbb{Z})\\ \int_\beta\mathrm{c}_1=d}} \left\langle[\mathrm{pt}]\,\psi^{d-2}\right\rangle^X_{0,1,\beta} \quad(d\geq2). \end{gathered} \end{equation}$$It is a deformation invariant of $X$. See Coates–Corti–Galkin–Kasprzyk, §B for the definition.
- Degrees and coefficients
- $d$ is the curve's anticanonical degree, with $\mathrm{c}_1=\mathrm{c}_1(\mathrm{T}_X)$.
- Showing more terms
- Click the period to add six coefficients. Where a quantum differential system is available, its recurrence computes further terms exactly; otherwise expansion stops at the recorded prefix.
- Quantum differential equation
- The scalar operator $L_X=\sum_m p_m(t)D^m$, with $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, annihilates the regularized period: $L_X\widehat{\mathrm{G}}_X=0$. Its polynomial coefficients are printed to the left of powers of $D$, as in Coates–Galkin–Kasprzyk–Strangeway, Appendix B.
- Period table and search
- The period table lists the same twelve initial coefficients for each family with recorded data. The search accepts individual coefficient constraints; the overview displays them in degree order.
Previously recorded quantum periods
Each family page compares its regularized quantum period with three published catalogues:
- Kalashnikov (2019), Four-dimensional Fano quiver flag zero loci: 749 entries, numbered CKK.
- Coates–Galkin–Kasprzyk–Strangeway (2020), Quantum periods for certain four-dimensional Fano manifolds: 272 named table entries.
- Coates–Kasprzyk–Prince (2015), Four-dimensional Fano toric complete intersections: 738 entries, numbered CKP.
The comparison uses exact coefficients in anticanonical degree, beginning at degree zero, with no rescaling. Named paper entries are cross-checked against the authors' longer coefficient lists in the database of regularized quantum periods. A match requires at least 16 coefficients and agreement of every overlapping available coefficient; hovering over an entry shows how many were compared.
“No match” means that every entry in that catalogue disagrees with an available coefficient. “Insufficient coefficients” means a compatible prefix exists but fewer than 16 coefficients can be compared. Agreement of finite prefixes does not prove equality of infinite series or an identification of deformation families. Absence from these three catalogues does not establish absence from all literature.