nLab quantum cohomology ring

Idea

Certain topological string theories (2d topological sigma-models [Witten 1988]), with target space a suitable symplectic manifold $X$, have as spaces of quantum states the ordinary cohomology (real/complex/de Rham cohomology) $H^\bullet(X)$ of the target manifold, but such that the genus$=0$ worldsheet 3-point function (a Gromov-Ruan-Witten invariant) equips the underlying vector space of the cohomology groups with a new product $H^\bullet(X) \otimes H^\bullet(X) \to H^\bullet(X)$ 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.

Examples

Example

(quantum cohomology ring of complex projective space)
The (small) quantum cohomology ring of complex projective space $\mathbb{C}P^{N-1}$, $N \geq 2$, should be

$$QH^\bullet\big( \mathbb{C}P^{N-1} ;\, \mathbb{C} \big) \;\simeq\; \mathbb{C}\big[ a_2,\, b_{2N} \big]/(a_2^N - b_{2N}) \,.$$

and as such a “deformation” of the ordinary cohomology ring, which is (see there):

$$H^\bullet\big( \mathbb{C}P^{N-1} ;\, \mathbb{C} \big) \;\simeq\; \mathbb{C}\big[ a_2 \big]/(a_2^N) \,.$$

(Witten 1990, p. 275; Cecotti & Vafa 1992, p. 3; Cecotti & Vafa 1993 p. 89; Bourdeau & Douglas 1994 3.4; see also Dorfmeister, Guest & Rossman 2010, p .1)

Related concepts

• Quantum cohomology arises as chiral rings in super Yang-Mills theory.

• Pontryagin ring

References

Quantum cohomology rings

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

• Edward Witten, Topological sigma models, Comm. Math. Phys. 118 3 (1988) 411-449 [euclid:cmp/1104162092]

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 $\mathbb{C}P^1$)

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

• Bernd Siebert, Gang Tian, On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator, Asian Journal of Mathematics 1 4 (1997) 679 – 695 [arXiv:alg-geom/9403010, doi:10.4310/AJM.1997.v1.n4.a2]

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:

• Maxim Kontsevich, On the History of quantum cohomology and homological mirror symmetry (2021) [video:YT]

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:

• Wikipedia, Quantum cohomology

Relation to chiral algebras:

• Fyodor Malikov, Vadim Schechtman, Deformations of chiral algebras and quantum cohomology of toric varieties [arXiv:math/0001170]

Slides of a talk for an audience of mathematical string theorists are

• Eric Sharpe,

  Quantum sheaf cohomology (pdf) Brandeis university (2010)

  Quantum sheaf cohomology I (pdf)

Of flag varieties

On quantum cohomology rings of flag varieties — “quantum Schubert calculus”:

• Dale H. Peterson (notes by Arun Ram), Quantum Cohomology of $G/H$, MIT (1997) [web, pdf, pdf]

• Naichung Conan Leung, Changzheng Li, An update of quantum cohomology of homogeneous varieties [arXiv:1407.5905]

(e.g. for $\mathbb{C}P^1 = SL(2,\mathbb{C})/P$)

Specifically the case of $\mathbb{C}P^N$ ($\mathbb{C}P^N$ 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 $N=2$ 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 $N$ $\mathbb{C}P^N$ 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 $\mathbb{C}P^1$:

• Josef F. Dorfmeister, Martin A. Guest, Wayne Rossman, The $t t^\ast$ Structure of the Quantum Cohomology of $\mathbb{C}P^1$ from the Viewpoint of Differential Geometry, Asian Journal of Mathematics 14 3 (2010) 417-438 [doi:10.4310/AJM.2010.v14.n3.a7]

Quantum K-cohomology rings

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 $\overline{\mathcal{M}}_{0,n}/S_n$ [arXiv:1508.02690]

  II. Fixed point localization [arXiv:1508.04374]

  III. Lefschetz’ formula on $\overline{\mathcal{M}}_{0,n}/S_n$ and adelic characterization [arXiv:1508.06697]

  IV. $D_q$-modules [arXiv:1508.06697]

  V. Toric $q$-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:

• Anders S. Buch, Leonardo C. Mihalcea, Quantum K-theory of Grassmannians, Duke Math. J. 156 3 (2011) 501-538 [arXiv:0810.0981, doi:10.1215/00127094-2010-218]

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 $\mathcal{N}=2$ SQCD and the quantum K-theory of the Grassmannian [arXiv:2309.06980]

More references are listed in:

• W. Gu, Leonardo C. Mihalcea, Eric Sharpe, H. Zou, Quantum K theory of symplectic Grassmannians [arXiv:2008.04909

See also:

• Wei Gu, Jirui Guo, Leonardo Mihalcea, Yaoxiong Wen, Xiaohan Yan, A correspondence between the quantum K theory and quantum cohomology of Grassmannians [arXiv:2406.13739]

Quantum cohomology as Pontrjagin rings

On the relation between quantum cohomology rings, hence of Gromov-Witten invariants in topological string theory, for flag manifold target spaces (such as the $\mathbb{C}P^1$-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 $K$ is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to

• Dale H. Peterson (notes by Arun Ram), Quantum Cohomology of $G/H$, MIT (1997) $[$web, pdf, pdf$]$

see also

• Yasha Savelyev, Quantum characteristic classes and the Hofer metric, Geom. Topol. 12 (2008) 2277-2326 $[$arXiv:0709.4510, doi:10.2140/gt.2008.12.2277$]$

and proven in

• Thomas Lam, Mark Shimozono, §6.2 in: Quantum cohomology of $G/P$ 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 $G/P$ $[$arXiv:2210.17382$]$

reviewed in

• Jimmy Chow, Homology of based loop groups and quantum cohomology of flag varieties, talk at Western Hemisphere Virtual Symplectic Seminar (2021) $[$pdf, pdf, video:YT$]$

with further discussion in:

• Naichung Conan Leung, Changzheng Li, Gromov-Witten invariants for $G/B$ and Pontryagin product for $\Omega K$, Transactions of the American Mathematical Society 364 5 (2012) 2567-2599 $[$web, arXiv:0810.4859, jstor:41524936$]$

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 $K$-groups of partial flag manifolds $[$arXiv:1906.09343$]$

• Syu Kato, Darboux coordinates on the BFM spaces $[$arXiv:2008.01310$]$

• Syu Kato, Quantum $K$-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:

• Constantin Teleman, Gauge theory and mirror symmetry $[$arXiv:1404.6305$]$

Relation to chiral rings of D=3 N=4 super Yang-Mills theory:

• Constantin Teleman, Coulomb branches for quaternionic representations $[$arXiv:2209.01088$]$