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Quantum \(K\)-theory. I: Foundations.(English)Zbl 1051.14064

The author studies the foundations of quantum \(K\)-theory, a \(K\)-theoretic version of quantum cohomology theory, giving a deformation of the ordinary \(K\)-ring \(K(X)\) of a smooth projective variety \(X\), analogous to the relation between quantum cohomology and ordinary cohomology. As a result, he develops a new class of Frobenius manifolds, and answers an open question ofA. Bayer andY. I. Manin [http://arxiv.org/abs/math.AG/0103164].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
19E08 \(K\)-theory of schemes
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
55N15 Topological \(K\)-theory

Cite

References:

[1]P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann-Roch , Séminaire de Geometrie Algébrique du Bois-Marie (SGA6), Lecture Notes in Math. 225 , Springer, Berlin, 1971. ·Zbl 0218.14001
[2]P. Baum, W. Fulton, and R. MacPherson, Riemann-Roch and topological \(K\) theory for singular varieties , Acta Math. 143 (1979), 155–192. ·Zbl 0474.14004 ·doi:10.1007/BF02392091
[3]A. Bayer and Yu. Manin, (Semi)simple exercises in quantum cohomology , ·Zbl 1077.14082
[4]K. Behrend, Gromov-Witten invariants in algebraic geometry , Invent. Math. 127 (1997), 601–617. ·Zbl 0909.14007 ·doi:10.1007/s002220050132
[5]K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. 128 (1997), 45–88. ·Zbl 0909.14006 ·doi:10.1007/s002220050136
[6]K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants , Duke Math. J. 85 (1996), 1–60. ·Zbl 0872.14019 ·doi:10.1215/S0012-7094-96-08501-4
[7]W. Fulton, Intersection Theory , 2nd ed., Ergeb. Math. Grenzgeb. (3) 2 , Springer, Berlin, 1998. ·Zbl 0541.14005
[8]W. Fulton and S. Lang, Riemann-Roch Algebra , Grundlehren Math. Wiss. 277 , Springer, New York, 1985. ·Zbl 0579.14011
[9]W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces , Mem. Amer. Math. Soc. 31 (1981), no. 243. ·Zbl 0467.55005
[10]W. Fulton and R. Pandharipande, ”Notes on stable maps and quantum cohomology” in Algebraic Geometry (Santa Cruz, Calif., 1995) , Proc. Sympos. Pure. Math. 62 , Part 2, Amer. Math. Soc., Providence, 1997, 45–96. ·Zbl 0898.14018
[11]A. Givental, Equivariant Gromov-Witten invariants , Internat. Math. Res. Notices 1996 , no. 13, 613–663. ·Zbl 0881.55006 ·doi:10.1155/S1073792896000414
[12]–. –. –. –., On the WDVV equation in quantum \(K\)-theory , Michigan Math. J. 48 (2000), 295–304. ·Zbl 1081.14523 ·doi:10.1307/mmj/1030132720
[13]A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices , Comm. Math. Phys. 168 (1995), 609–641. ·Zbl 0828.55004 ·doi:10.1007/BF02101846
[14]A. Givental and Y.-P. Lee, Quantum \(K\)-theory on flag manifolds, finite-difference Toda lattices and quantum groups , Invent. Math. 151 (2003), 193–219. ·Zbl 1051.14063 ·doi:10.1007/s00222-002-0250-y
[15]T. Graber and R. Pandharipande, Localization of virtual classes , Invent. Math. 135 (1999), 487–518. ·Zbl 0953.14035 ·doi:10.1007/s002220050293
[16]B. Kim, Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices , Ann. of Math. (2) 149 (1999), 129–148. JSTOR: ·Zbl 1054.14533 ·doi:10.2307/121021
[17]M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Comm. Math. Phys. 147 (1992), 1–23. ·Zbl 0756.35081 ·doi:10.1007/BF02099526
[18]–. –. –. –., ”Enumeration of rational curves via torus actions” in The Moduli Space of Curves (Texel Island, Netherlands, 1994) , Progr. Math. 129 , Birkhäuser, Boston, 1995, 335–368.
[19]M. Kontsevich and Yu. Manin, Gromov-Witten Classes, quantum cohomology, and enumerative geometry , Comm. Math. Phys. 164 (1994), 525–562. ·Zbl 0853.14020 ·doi:10.1007/BF02101490
[20]A. Kresch, Canonical rational equivalence of intersections of divisors , Invent. Math. 136 (1999), 483–496. ·Zbl 0923.14003 ·doi:10.1007/s002220050317
[21]Y.-P. Lee, A Formula for Euler characteristics of tautological line bundles on the Deligne-Mumford moduli spaces , Internat. Math. Res. Notices 1997 , no. 8, 393–400. ·Zbl 0949.14016 ·doi:10.1155/S1073792897000263
[22]——–, Orbifold Euler characteristics of universal cotangent line bundles on \(\overlineM_1,n\) , preprint, 2003,
[23]——–, Quantum \(K\)-theory, II: Computations and open problems , in preparation.
[24]Y.-P. Lee and R. Pandharipande, A reconstruction theorem in quantum cohomology and quantum \(K\)-theory , ·Zbl 1080.14065 ·doi:10.1353/ajm.2004.0049
[25]J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties , J. Amer. Math. Soc. 11 (1998), 119–174. JSTOR: ·Zbl 0912.14004 ·doi:10.1090/S0894-0347-98-00250-1
[26]Yu. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces , Amer. Math. Soc. Colloq. Publ. 47 , Amer. Math. Soc., Providence, 1999. ·Zbl 0952.14032
[27]J. Morava, ”Quantum generalized cohomology” in Operads: Proceedings of Renaissance Conferences (Hartford, Conn./Luminy, France, 1995) , Contemp. Math. 202 , Amer. Math. Soc., Providence, 1997, 407–419. ·Zbl 1004.55002
[28]E. Witten, ”Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, Mass., 1990) , Amer. Math. Soc., Providence, 1991, 243–310. ·Zbl 0757.53049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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