Non-toric Brane Webs, Calabi–Yau 3-Folds, and 5d SCFTs

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 $$\mathbb {R}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> with focus-focus singularities introduced. To any such a 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 $$\mathcal {X}^{\textrm{can}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> <mml:mtext>can</mml:mtext> </mml:msup> </mml:math> to ( Y , D , L ), as a non-toric canonical 3-fold singularity, and show that M-theory on $$\mathcal {X}^{\textrm{can}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> <mml:mtext>can</mml:mtext> </mml:msup> </mml:math> engineers the same 5d SCFT as W . We also explain how to derive explicit equations for $$\mathcal {X}^{\textrm{can}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> <mml:mtext>can</mml:mtext> </mml:msup> </mml:math> using scattering diagrams, encoding disk worldsheet instantons in the A-model, or equivalently the BPS states of an auxiliary rank one 4d $$\mathcal {N}=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> theory.