Search for: All records

Abstract We study webs of 5-branes with 7-branes in Type IIB string theory from a geometric perspective. Mathematically, a web of 5-branes with 7-branes is a tropical curve in$$\mathbb {R}^2$$ $R^2$with focus-focus singularities introduced. To any such a webW, we attach a log Calabi–Yau surface (Y, D) with a line bundleL. We then describe supersymmetric webs, which are webs defining 5d superconformal field theories (SCFTs), in terms of the geometry of (Y, D, L). We also introduce particular supersymmetric webs called “consistent webs, and show that any 5d SCFT defined by a supersymmetric web can be obtained from a consistent web by adding free hypermultiplets. Using birational geometry of degenerations of log Calabi–Yau surfaces, we provide an algorithm to test the consistency of a web in terms of its dual polygon. Moreover, for a consistent webW, we provide an algebro-geometric construction of the mirror$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$to (Y, D, L), as a non-toric canonical 3-fold singularity, and show that M-theory on$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$engineers the same 5d SCFT asW. We also explain how to derive explicit equations for$$\mathcal {X}^{\textrm{can}}$$ $X^{\text{can}}$using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d$$\mathcal {N}=2$$ $N=2$theory. 

We prove a correspondence between Donaldson–Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov–Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram, describing the wall-crossing behavior of Donaldson–Thomas invariants, with a scattering diagram capturing punctured Gromov–Witten invariants via tropical geometry. 

Abstract Cluster varieties come in pairs: for any $\mathcal{𝒳}${\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{𝒳}${\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.