Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs

Journal article

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 $\mathbb{R}^2$ with focus-focus singularities introduced. To any such web $W$, we attach a log Calabi--Yau surface $(Y, D)$ with a line bundle $L$. 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 web $W$, we provide an algebro-geometric construction of the mirror $X^{\mathrm{can}}$ to $(Y, D, L)$, as a non-toric canonical threefold singularity, and show that M-theory on $X^{\mathrm{can}}$ engineers the same 5d SCFT as $W$. We also explain how to derive explicit equations for $X^{\mathrm{can}}$ using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank-one 4d $\mathcal{N}=2$ theory.

Authors: Alexeev, V; Arguz, H; Bousseau, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Nature
• Journal: Communications in Mathematical Physics
• Volume: 406
• Issue: 11
• Article number: 287
• Publication date: 2025-10-09
• Acceptance date: 2025-09-03
• DOI: 10.1007/s00220-025-05461-9
• EISSN: 1432-0916
• ISSN: 0010-3616
• Language: English