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Floer Theory of Higher Rank Quiver 3-folds

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Peer-reviewed

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Article

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Abstract

jats:titleAbstract</jats:title>jats:pWe study threefolds jats:italicY</jats:italic> fibred by jats:inline-formulajats:alternativesjats:tex-math$$A_m$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miA</mml:mi> mml:mim</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>-surfaces over a curve jats:italicS</jats:italic> of positive genus. An ideal triangulation of jats:italicS</jats:italic> defines, for each rank jats:italicm</jats:italic>, a quiver jats:inline-formulajats:alternativesjats:tex-math$$Q(\Delta m)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miQ</mml:mi> mml:mo(</mml:mo> mml:msub mml:miΔ</mml:mi> mml:mim</mml:mi> </mml:msub> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, hence a jats:inline-formulajats:alternativesjats:tex-math$$CY_3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miC</mml:mi> mml:msub mml:miY</mml:mi> mml:mn3</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-category jats:inline-formulajats:alternativesjats:tex-math$$\mathcal {C}(W)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miC</mml:mi> mml:mo(</mml:mo> mml:miW</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for any potential jats:italicW</jats:italic> on jats:inline-formulajats:alternativesjats:tex-math$$Q(\Delta m)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miQ</mml:mi> mml:mo(</mml:mo> mml:msub mml:miΔ</mml:mi> mml:mim</mml:mi> </mml:msub> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. We show that for jats:inline-formulajats:alternativesjats:tex-math$$\omega $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miω</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of jats:inline-formulajats:alternativesjats:tex-math$$(Y,\omega )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mo(</mml:mo> mml:miY</mml:mi> mml:mo,</mml:mo> mml:miω</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is quasi-isomorphic to jats:inline-formulajats:alternativesjats:tex-math$$(\mathcal {C},W{[\omega ]})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mo(</mml:mo> mml:miC</mml:mi> mml:mo,</mml:mo> mml:msub mml:miW</mml:mi> mml:mrow mml:mo[</mml:mo> mml:miω</mml:mi> mml:mo]</mml:mo> </mml:mrow> </mml:msub> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for a certain generic potential jats:inline-formulajats:alternativesjats:tex-math$$W{[\omega ]}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miW</mml:mi> mml:mrow mml:mo[</mml:mo> mml:miω</mml:mi> mml:mo]</mml:mo> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>. 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 jats:inline-formulajats:alternativesjats:tex-math$${\mathbb {P}}GL_{m+1}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miP</mml:mi> mml:miG</mml:mi> mml:msub mml:miL</mml:mi> mml:mrow mml:mim</mml:mi> mml:mo+</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>-local systems on jats:italicS</jats:italic>, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {S}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miS</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>’.</jats:p>

Description

Keywords

4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

Communications in Mathematical Physics

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

Publisher

Springer Science and Business Media LLC
Sponsorship
Engineering and Physical Sciences Research Council (EP/N01815X/1)