group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Certain topological string theories (2d topological sigma-models [Witten 1988]), with target space a suitable symplectic manifold , have as spaces of quantum states the ordinary cohomology (real/complex/de Rham cohomology) of the target manifold, but such that the genus worldsheet 3-point function (a Gromov-Ruan-Witten invariant) equips the underlying vector space of the cohomology groups with a new product that deforms (“quantizes”) the ordinary cup product/wedge product-cohomology ring to a non-commutative ring [Witten (1990), §3], whence called a quantum cohomology ring [Vafa (1992)].
Beware that later authors often abbreviate the term from quantum cohomology ring to just quantum cohomology, which is however a misnomer since it is not the underlying notion of (ordinary) cohomology that is being deformed/quantized here (just the coefficients are typically enlarged), but the multiplicative ring-structure on cohomology is deformed – indeed the title of Vafa (1992) speaks more properly of quantum rings.
(quantum cohomology ring of complex projective space)
The (small) quantum cohomology ring of complex projective space , , should be
and as such a “deformation” of the ordinary cohomology ring, which is (see there):
Quantum cohomology arises as chiral rings in super Yang-Mills theory.
The notion of quantum cohomology originates as a model for certain topological string n-point functions in:
Cumrun Vafa, §4 in: Topological mirrors and quantum rings, in Shing-Tung Yau (ed.) Essays on mirror manifolds, International Press (1992), republished in Mirror Symmetry I, AMS/IP Studies in Advanced Mathematics 9 (1998) [arXiv:hep-th/9111017, doi:10.1090/amsip/009]
Edward Witten, §3 in: Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry 1 (1990) 243-310 [doi:10.4310/SDG.1990.v1.n1.a5, inspire:307956, SemanticScholar]
motivated by
See also:
Eric Zaslow, Topological Orbifold Models and Quantum Cohomology Rings, Commun. Math. Phys. 156 (1993) 301-331 [arXiv:hep-th/9211119, doi:10.1007/BF02098485]
(targets orbifolds of )
The rigorous mathematical formulation in differential symplectic geometry via Gromov-Ruan-Witten invariants is due to:
Yongbin Ruan, Gang Tian, A mathematical theory of quantum cohomology, Mathematical Research Letters 1 2 (1994) 269-278 [doi:10.4310/MRL.1994.v1.n2.a15, pdf]
Yongbin Ruan, Gang Tian, A mathematical theory of quantum cohomology, J. Diff. Geometry 42 2 (1995) 259-367 [doi:10.4310/jdg/1214457234]
with early computations in
and in terms of algebraic geometry and via Frobenius manifolds due to:
Maxim Kontsevich, Yuri Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164 (1994) 525-562 [doi:10.1007/BF02101490, hep-th/9402147]
Maxim Kontsevich, Yuri Manin, Ralph Kaufmann, Quantum cohomology of a product, Invent. Math. 124 (1996) 313-339 [doi:10.1007/s002220050055 arXiv:q-alg/9502009]
Yuri I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, Colloquium Publications 47 (1999) [ISBN:978-0-8218-1917-3, mpim-eprint:1598]
See also:
Boris Dubrovin, Geometry and Analytic Theory of Frobenius Manifolds, Extra Volume ICM II (1998) 315–326 [arXiv:math/9807034]
Ron Donagi, Josh Guffin, Sheldon Katz, Eric Sharpe, A Mathematical Theory of Quantum Sheaf Cohomology, Asian J. Math. 18 (2014) 387-418 [arXiv:1110.3751, doi:10.4310/AJM.2014.v18.n3.a1]
More on the history:
Introduction and review:
Martin A. Guest, Introduction to Quantum Cohomology, Vietnam Journal of Mathematics 33 SI (2005) 29–59 [pdf]
Joachim Kock, Israel Vainsencher, An Invitation to Quantum Cohomology – Kontsevich’s Formula for Rational Plane Curves, Birkhäser (2007) [doi:10.1007/978-0-8176-4495-6]
Martin A. Guest, Quantum cohomology: is it still relevant? [arXiv:2210.05413]
Josh Guffin, Quantum sheaf cohomology, a précis, Matemática Contemporânea 41 (2012) 17-26 [arxiv:1101.1305]
Tom Coates, An Introduction to Quantum Cohomology [pdf, pdf]
Alexander Givental, A tutorial on Quantum Cohomology [pdf, pdf]
See also:
Relation to chiral algebras:
Slides of a talk for an audience of mathematical string theorists are
On quantum cohomology rings of flag varieties — “quantum Schubert calculus”:
Dale H. Peterson (notes by Arun Ram), Quantum Cohomology of , MIT (1997) [web, pdf, pdf]
Naichung Conan Leung, Changzheng Li, An update of quantum cohomology of homogeneous varieties [arXiv:1407.5905]
(e.g. for )
Specifically the case of ( sigma-model):
Edward Witten, On the structure of the topological phase of two-dimensional gravity, Nuclear Physics B 340 2–3 (1990) 281-332 [doi:10.1016/0550-3213(90)90449-N]
Sergio Cecotti, Cumrun Vafa, Exact Results for Supersymmetric Sigma Models, Phys. Rev. Lett. 68 (1992) 903-906 [arXiv:hep-th/9111016, doi:10.1103/PhysRevLett.68.903]
Sergio Cecotti, Cumrun Vafa, On Classification of Supersymmetric Theories, Comm. Math. Phys. 158 (1993) 569-644 [arXiv:hep-th/9211097, doi:10.1007/BF02096804]
M. F. Bourdeau, Michael R. Douglas, Topological-Antitopological Fusion and the Large Model, Nucl. Phys. B 420 (1994) 243-267 [doi:hep-th/9312095, doi:10.1016/0550-3213(94)90380-8]
And specifically the case of :
The idea of an analogous quantum deformation of topological K-theory-rings originates around:
Alexander B. Givental, On the WDVV-equation in quantum K-theory, Michigan Math. J. 48 1 (2000) 295-304 [arXiv:math/0003158, doi:10.1307/mmj/1030132720]
Y.-P. Lee, Quantum K-Theory I: Foundations, Duke Math. J. 121 3 (2004) 389-424 [arXiv:math/0105014, doi:10.1215/S0012-7094-04-12131-1]
and for some form of equivariant K-theory in:
Alexander B. Givental, Permutation-equivariant quantum K-theory [webpage]
I. Definitions. Elementary K-theory of [arXiv:1508.02690]
II. Fixed point localization [arXiv:1508.04374]
III. Lefschetz’ formula on and adelic characterization [arXiv:1508.06697]
IV. -modules [arXiv:1508.06697]
V. Toric -hypergeometric functions [arXiv:1509.03903]
VI. Mirrors [arXiv:1509.07852]
VII. General theory [arXiv:1510.03076]
VIII. Explicit reconstruction [arXiv:1510.06116]
IX. Quantum Hirzebruch-Riemann-Roch in all genera [arXiv:1709.03180]
X. Quantum Hirzebruch-Riemann-Roch in genus 0, SIGMA 16 031 (2020) [arXiv:1710.02376, doi:10.3842/SIGMA.2020.031]
XI. Quantum Adams-Riemann-Roch [arXiv:1711.04201]
See also:
Relation to D=3 N=2 super Yang-Mills theory:
Hans Jockers, Peter Mayr, Quantum K-Theory of Calabi-Yau Manifolds, J. High Energ. Phys. 2019 11 (2019) [arXiv:1905.03548, doi:10.1007/JHEP11(2019)011]
Hans Jockers, Peter Mayr, A 3d Gauge Theory/Quantum K-Theory Correspondence, Advances in Theoretical and Mathematical Physics 24 2 (2020) [arXiv:1808.02040, doi:10.4310/ATMP.2020.v24.n2.a4]
Cyril Closset, Osama Khlaif, Grothendieck lines in 3d SQCD and the quantum K-theory of the Grassmannian [arXiv:2309.06980]
More references are listed in:
See also:
On the relation between quantum cohomology rings, hence of Gromov-Witten invariants in topological string theory, for flag manifold target spaces (such as the -sigma model) and Pontrjagin rings (homology-Hopf algebras of based loop spaces):
That the Pontryagin ring-structure on the ordinary homology of the based loop space of a simply-connected compact Lie group is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to
see also
and proven in
Thomas Lam, Mark Shimozono, §6.2 in: Quantum cohomology of and homology of affine Grassmannian, Acta Mathematica 204 (2010) 49–90 arXiv:0705.1386, doi:10.1007/s11511-010-0045-8
Chi Hong Chow, Peterson-Lam-Shimozono’s theorem is an affine analogue of quantum Chevalley formula arXiv:2110.09985
Chi Hong Chow, On D. Peterson’s presentation of quantum cohomology of arXiv:2210.17382
reviewed in
with further discussion in:
On the variant for Pontryagin products not on ordinary homology but in topological K-homology:
Thomas Lam, Changzheng Li, Leonardo C. Mihalcea, Mark Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of Algebra 513 (2018) 326-343 doi:10.1016/j.jalgebra.2018.07.029, arXiv:1705.03435
Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno, Peterson Isomorphism in K-theory and Relativistic Toda Lattice, International Mathematics Research Notices 2020 19 (2020) 6421–6462 arXiv:1703.08664, doi:10.1093/imrn/rny051
Syu Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds arXiv:1805.01718
Syu Kato, On quantum -groups of partial flag manifolds arXiv:1906.09343
Syu Kato, Darboux coordinates on the BFM spaces arXiv:2008.01310
Syu Kato, Quantum -groups on flag manifolds, talk at IMPANGA 20 (2021) pdf
On the example of the CP^1 sigma-model: LLMS18, §4.1, Kato21 p. 17, Chow22 Exp. 1.4.
See also:
Relation to chiral rings of D=3 N=4 super Yang-Mills theory:
Last revised on June 21, 2024 at 07:48:53. See the history of this page for a list of all contributions to it.