81 problems

Let $X$ be a smooth algebraic variety whose bounded derived category of coherent sheaves is equipped with a Frobenius structure, and let $D^b\operatorname{Coh}(X)$ denote that boun…

The WDVV equations are the associativity equations for a Frobenius manifold prepotential, and a polynomial WDVV solution means a polynomial prepotential satisfying these equations.…

Let $M$ be a projective manifold whose genus $0$ Gromov–Witten potential is semisimple, and let ${\mathscr D}_{Giv}$ be the formal total potential constructed from it by Givental's…

Central-invariant conjecture. The central invariants of the bihamiltonian structure of the $(n,1)$-type rational reduction of the 2D-Toda hierarchy are all equal to

Let $H_{0;1,N-1}$ be the Hurwitz space of rational functions on the Riemann sphere with two poles, one of order $1$ and one of order $N-1$, where $N\geq 2$. Each such space carries…

Let the Principal Hierarchy be associated to a semisimple Frobenius manifold, and consider a deformation of its bihamiltonian structure with constant central invariants. Denote the…

A WDVV solution is a solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations, and it has the required good analytic properties when it satisfies the analytic conditions intend…

Genus two G-function vanishing conjecture. For every such $M$,

A Frobenius manifold constructed from a Coxeter group $W$ has Euler vector field … where the $d_i$ are the exponents, equivalently the degrees of the basic $W$-invariant polynomial…

Let $M$ be a Calabi–Yau manifold. The Dolbeault cohomology $A(M)=H^{*,*}(M)$ carries a formal Frobenius manifold structure arising from the DGBV algebra…

Let $\mathsf{H}(V)$ be the Frobenius manifold associated with a smooth projective variety $V$, and assume that it is semisimple. Let $\{\!\{u_a(x),u_b(y)\}\!\}_0$ and…

Let $V$ be a smooth projective variety, and let $\mathsf{H}(V)$ be its small phase space, regarded as a Frobenius manifold determined by the genus-$0$ Gromov–Witten invariants. Ass…

The elliptic-potential conjecture. The elliptic Gromov–Witten potential of $X$ is determined by

The generalized mirror conjecture. The elliptic Gromov–Witten invariant $dG$ is expressed by this formula in terms of rational Gromov–Witten invariants. This is the proposed bridge…

The primitive-form conjecture. The one-form $dG$ satisfies all axioms for the genus-one Gromov–Witten invariant; in particular, Getzler's relation holds for $dG$. This proposes a s…

Barannikov–Kontsevich conjecture. The Frobenius structure $(S',\eta_F|_{S'},\Phi_F|_{S'})$ should be a Frobenius submanifold of the Frobenius manifold constructed by Barannikov–Kon…

Barannikov–Kontsevich mirror symmetry conjecture. The resulting formal Frobenius manifold of the $B$-model can be identified with the quantum cohomology of the mirror dual Calabi–Y…

Let $\mathcal{X}$ be the orbifold and $Y$ its crepant resolution, with big quantum products parametrized respectively by $\tau\in H(\mathcal{X})$ and $t\in H(Y)$. Together with the…

Let $\mathcal{X}$ be an orbifold with projective coarse moduli space $X$, and let $Y$ be a crepant resolution. Consider the quantum cohomology Frobenius manifolds associated to…

Twisted equivariant abelian/nonabelian correspondence. Let $\varphi:P'\to N'$ be the isomorphism of formal schemes over…

Equivariant abelian/nonabelian correspondence. Let $\varphi:P\longrightarrow N$ be the isomorphism of formal schemes over $N(X//{\bf G})\otimes{\mathbb C}[\lambda]$ defined by…

The abelian/nonabelian quantum-product conjecture. There is an equality

Let $M$ be the base space of an unfolding $F$ of a singularity with a CDV-structure, let $R\subset M$ be the real analytic exceptional subvariety, let $E+\overline E$ be the associ…

Let a Frobenius manifold be described by a semi-infinite subspace $\mathcal{L}(x)\in Gr_{\frac{\infty}{2}}^{(n)}$, with higher-time parameters $\{x^{\alpha,r}\}$ for…

Analytic extension conjecture. The higher-genus potentials produced by formula (1) should extend analytically through the bifurcation set; consequently, they must vanish for singul…