2020/02/25 by Ivan Smith
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Brane #Combinatorics #Conjecture #Fibered knot #Genus #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematical physics #Mathematics #Omega #Physics #Pure mathematics #Quiver #Rank (graph theory) #Sign (mathematics) #Subcategory #Symplectic geometry #math-ph #math.AG #math.MP #math.SG #msc:14J32 #msc:53D37
paper · pdf · doi:10.1007/s00220-021-04252-2
published in Communications in Mathematical Physics 388(3), 1181-1203 (Springer Science+Business Media) · 23 pages, 14 figures
arxiv created 2020/02/25 · openalex publication_date 2021/11/11 · arxiv updated 2021/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We study threefolds Y fibred by Am <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m , a quiver Q(Δ m) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , hence a CY3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> -category \mathcal C(W) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>W</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for any potential W on Q(Δ m) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We show that for ω <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of (Y,ω ) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is quasi-isomorphic to (\mathcal C,W[ω ]) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for a certain generic potential W[ω ] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed \mathbb PGLm+1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> -local systems on S , and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class \mathcal S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> ’.