03443nam a22005175i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001400153072001700167072002300184072001600207100007800223245014200301250001800443264006900461300006600530336002600596337002600622338003600648347002400684490005300708505029400761520120101055650002402256650002602280650002802306650002802334650002602362650002402388650002602412650002802438650002702466650004202493700008002535700007902615710003402694773002602728776003602754776003602790830005302826856004602879978-3-031-69067-9DE-He21320251030113250.0cr nn 008mamaa241029s2024    sz |    s    |||| 0|eng d  a97830316906797 a10.1007/978-3-031-69067-92doi 4aQA564-609 7aPBMW2bicssc 7aMAT0120102bisacsh 7aPBMW2thema1 aCotti, Giordano.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aHelix Structures in Quantum Cohomology of Fano Varietiesh[electronic resource] /cby Giordano Cotti, Boris A. Dubrovin, Davide Guzzetti.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXIII, 236 p. 18 illus., 15 illus. in color.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aLecture Notes in Mathematics,x1617-9692 ;v23560 a- Introduction -- Gromov–Witten Theory and Quantum Cohomology -- Helix Theory in Triangulated Categories -- Non-Symmetric Orthogonal Geometry of Mukai Lattices -- The Main Conjecture -- Proof of the Main Conjecture for Projective Spaces -- Proof of the Main Conjecture for Grassmannians.  aThis research monograph provides a comprehensive study of a conjecture initially proposed by the second author at the 1998 International Congress of Mathematicians (ICM). This conjecture asserts the equivalence, for a Fano variety, between the semisimplicity condition of its quantum cohomology and the existence of full exceptional collections in its derived category of coherent sheaves. Additionally, in its quantitative form, the conjecture specifies an explicit relation between the monodromy data of the quantum cohomology, characteristic classes, and exceptional collections. A refined version of the conjecture is introduced, with a particular focus on the central connection matrix, and a precise link is established between this refined conjecture and Γ-conjecture II, as proposed by S. Galkin, V. Golyshev, and H. Iritani. By performing explicit calculations of the monodromy data, the validity of the refined conjecture for all complex Grassmannians G(r,k) is demonstrated. Intended for students and researchers, the book serves as an introduction to quantum cohomology and its isomonodromic approach, along with its algebraic counterpart in the derived category of coherent sheaves. 0aAlgebraic geometry. 0aMathematical physics. 0aDifferential equations. 0aGeometry, Differential. 0aAlgebra, Homological.14aAlgebraic Geometry.24aMathematical Physics.24aDifferential Equations.24aDifferential Geometry.24aCategory Theory, Homological Algebra.1 aDubrovin, Boris A.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut1 aGuzzetti, Davide.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut2 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303169066208iPrinted edition:z9783031690686 0aLecture Notes in Mathematics,x1617-9692 ;v235640uhttps://doi.org/10.1007/978-3-031-69067-9