28 problems

A torically constructed family of Calabi–Yau manifolds has a projective normal form Picard–Fuchs equation. A no-apparent-singularities conjecture. No apparent singularities arise i…

Mirror symmetry conjecture. The operators $L_{A_X}$ are of Picard–Fuchs type: there exist such a pencil $\pi$ and meromorphic section $\omega$ for which a period $\Phi$ satisfies

Let $({\cal M}_i,\omega_i)$, $i=1,2$, be pairs consisting of a family of algebraic varieties over $\mathbb{C}(t)$ and a differential form as above, with…

Let $z:=\phi^3$, let $\theta:=z\frac{d}{dz}$, and let $I_3$ be the logarithmic solution of the Picard–Fuchs equation described in the source. Define … where $I^{(0)}_2$ is the holo…

Let $X$ be a noncompact toric Calabi–Yau threefold with $\dim H_4(X,\mathbb{Z})=0$, and let $\{l^1,\dots,l^m\}$ define $X$ via symplectic quotient. Define … where … and … If…

Consider the A-model on a Calabi–Yau threefold $X$. Let $u_i$ be the logarithm of the B-model complex deformation parameter $z_i$ obtained from the toric construction of the mirror…

Consider the noncompact relative Picard–Fuchs system with moduli coordinates $z_i$, arising from the relative cohomology of the pair consisting of the hypersurface and its divisor.…

Let $\Lambda=\{l\in\mathbb{Z}^{n+1}:\sum_{i=0}^n l_i m_i=0,\ \sum_{i=0}^n l_i=0\}$ be the lattice of relations among the exponent vectors of the Laurent polynomial defining the tor…

Let a scattering problem in particle physics determine an appropriate moduli space and differential forms on it. A scattering amplitude, or correlation function, is expected to be…

Integral symplectic monodromy conjecture. The basis of periods has the property asserted in the continuation of the source statement.

Let $Q$ be a parameter and let $I_1(Q)$ and $I_2(Q)$ be periods satisfying the Picard–Fuchs equation in the source, with $I_1(Q)=\log Q+I_1^{\mathrm{hol}}(Q)$. Define the canonical…

Geometric-realisation conjecture. Every Calabi–Yau operator has a geometric realisation.

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $f:X\to S$ be a smooth proper morphism of smooth $R$-schemes, fix $s\in S(R)$, and let $\widehat{S}$ and…

For an $n$-loop banana amplitude in dimension $D=2$ with generic masses, let $\operatorname{ord}(n)$ denote the order of its Picard–Fuchs equation. Lairez's order conjecture. The o…

Let $\hat{E}_t^{(n)}$ be the Fourier-transformed differential operator associated with the $n$-loop banana amplitude, and let $\widehat{PF}^{(n)}_{t}$ be its Picard–Fuchs different…

Let $\mathcal{P}$ be an orphan Picard–Fuchs operator of order four, let $\mathcal{S}ol(\mathcal{P})$ be its space of solutions, and let $\mathcal{B}_{m}$ be the Frobenius basis at…

K3-period conjecture. The tardigrade Feynman integrals are period integrals of $K3$ surfaces; in the generic two-loop case, they are relative periods of a $K3$ surface of Picard nu…

Relative Calabi–Yau period conjecture. The sunset Feynman integrals are relative periods of Calabi–Yau varieties of dimension $n-2$.

Hauptmodul and monodromy-group conjecture. Then $k^n$ is a Hauptmodul for $\Gamma^b$,

Irreducibility conjecture. In the hyperoctahedral case, the Picard–Fuchs operator $L$ is irreducible in $\mathbb{Q}(t)[\theta]$ for all $n\ge2$.

Let $\mathcal{P}$ be a Picard–Fuchs operator of a family of Calabi–Yau threefolds with a MUM point at $0$, and suppose that its associated local system is irreducible. For a loop…

Let $\Pi_{P}$ be the canonical form of local solutions of the Picard–Fuchs differential equations at a special boundary point $P$, and suppose mirror symmetry arises from $P$. Inte…

Let $X$ be a Fano variety, let its $G$-series be the generating function for the relevant genus-zero Gromov–Witten invariants, and let a pencil of Calabi–Yau varieties have Picard–…

Let $f=x^p y^p(1-x-y)$, with $p\geq 1$, and let $V=H_1(f^{-1}(t),\mathbb Z)/V_0$ be the reduced first homology of a regular fiber, where $V_0$ is the kernel of the intersection for…

Degree and ramification conjecture. The following hold: