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dc.contributor.authorSmith, Ivan
dc.date.accessioned2021-10-21T23:30:32Z
dc.date.available2021-10-21T23:30:32Z
dc.date.issued2021-12
dc.identifier.issn0010-3616
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/329736
dc.description.abstract<jats:title>Abstract</jats:title><jats:p>We study threefolds <jats:italic>Y</jats:italic> fibred by <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_m$$</jats:tex-math><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></jats:alternatives></jats:inline-formula>-surfaces over a curve <jats:italic>S</jats:italic> of positive genus. An ideal triangulation of <jats:italic>S</jats:italic> defines, for each rank <jats:italic>m</jats:italic>, a quiver <jats:inline-formula><jats:alternatives><jats:tex-math>$$Q(\Delta _m)$$</jats:tex-math><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></jats:alternatives></jats:inline-formula>, hence a <jats:inline-formula><jats:alternatives><jats:tex-math>$$CY_3$$</jats:tex-math><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></jats:alternatives></jats:inline-formula>-category <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathcal {C}(W)$$</jats:tex-math><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></jats:alternatives></jats:inline-formula> for any potential <jats:italic>W</jats:italic> on <jats:inline-formula><jats:alternatives><jats:tex-math>$$Q(\Delta _m)$$</jats:tex-math><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></jats:alternatives></jats:inline-formula>. We show that for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\omega $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</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-formula><jats:alternatives><jats: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:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is quasi-isomorphic to <jats:inline-formula><jats:alternatives><jats: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: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></jats:alternatives></jats:inline-formula> for a certain generic potential <jats:inline-formula><jats:alternatives><jats:tex-math>$$W_{[\omega ]}$$</jats:tex-math><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></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-formula><jats:alternatives><jats:tex-math>$${\mathbb {P}}GL_{m+1}$$</jats:tex-math><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></jats:alternatives></jats:inline-formula>-local systems on <jats:italic>S</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-formula><jats:alternatives><jats:tex-math>$${\mathcal {S}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>’.</jats:p>
dc.publisherSpringer Science and Business Media LLC
dc.rightsAll rights reserved
dc.rights.urihttp://www.rioxx.net/licenses/all-rights-reserved
dc.titleFloer Theory of Higher Rank Quiver 3-folds
dc.typeArticle
prism.publicationNameCOMMUNICATIONS IN MATHEMATICAL PHYSICS
dc.identifier.doi10.17863/CAM.77183
dcterms.dateAccepted2021-10-17
rioxxterms.versionofrecord10.1007/s00220-021-04252-2
rioxxterms.versionAM
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.licenseref.startdate2021-10-17
dc.identifier.eissn1432-0916
rioxxterms.typeJournal Article/Review
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/N01815X/1)
cam.issuedOnline2021-11-11
cam.orpheus.successTue Feb 01 19:02:05 GMT 2022 - Embargo updated*
cam.orpheus.counter5
rioxxterms.freetoread.startdate2022-11-11


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