Related papers: Graph sums in the Remodeling Conjecture
For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space…
We obtain exact results in \alpha' for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yau's that we…
We prove the Landau-Ginzburg/Calabi-Yau correspondence between the Gromov-Witten theory of each elliptic orbifold curve and its Fan-Jarvis-Ruan-Witten theory counterpart via modularity. We show that the correlation functions in these two…
We prove the multiple cover formula conjecture for abelian surfaces for a large class of insertions, including all stationary invariants. The proof uses the reduced degeneration formula expressing the invariants in terms of the correlated…
We prove the KKV conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov-Witten/Pairs correspondence for K3-fibered…
Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces,…
To a pair $(A,s)$ consisting of a smooth, cyclic $A_\infty$-algebra $A$ and a splitting $s$ of the Hodge filtration on its Hochschild homology Costello (2005) associates an invariant which conjecturally generalizes the total descendant…
We consider Open Gromov-Witten invariants for noncompact Calabi-Yau in the case the Lagrangian has the topology of $\R^2 \times S^1$. The definition of the invariant involves the choice of a frame for the Lagrangian, in accord with string…
We provide a framework for abstract reconstruction problems using the $K$-theory of categories with covering families, which we then apply to reformulate the edge reconstruction conjecture in graph theory. Along the way, we state some…
We construct classes in the motivic cohomology of certain 1-parameter families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous…
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an…
We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We…
We prove the BCOV Feynman rule by identifying the Feynman graph sum to the graph sum of an R-matrix action extracted from the NMSP theory. As direct consequences, (i) we obtain the genus one and genus two potentials, and (ii) we prove the…
Toroidal 3-orbifolds $(S^1)^6/G$, for $G$ a finite group, were some of the earliest examples of Calabi-Yau 3-orbifolds to be studied in string theory. While much mathematical progress towards the predictions of string theory has been made…
The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge…
This paper develops a new chain model for the commutative graph complex $\mathsf{GC}_2$ which takes Lie graph homology as an input. Our main technical result is the identification of a large contractible complex of (certain) tadpoles and…
We show that, after the change of variables $q=e^{iu}$, refined floor diagrams for $\mathbb{P}^2$ and Hirzebruch surfaces compute generating series of higher genus relative Gromov-Witten invariants with insertion of a lambda class. The…
We derive conjectures, called genus 1 Enumerative mirror symmetry for moduli spaces of Higgs bundles, which relate curve-counting invariants of moduli spaces of Higgs $\mathrm{SL}_r$-bundles to curve-counting invariants of moduli spaces of…
We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. In this paper we construct the Open Gromov-Witten potential. The evaluation of the potential on its critical points leads to numerical invariants.
We prove an explicit formula for the genus-one Fan-Jarvis-Ruan-Witten invariants associated to the quintic threefold, verifying the genus-one mirror conjecture of Huang, Klemm, and Quackenbush. The proof involves two steps. The first step…