nLab quantum cohomology ring

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Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Quantum field theory

Contents

Idea

Certain topological string theories (2d topological sigma-models [Witten 1988]), with target space a suitable symplectic manifold XX, have as spaces of quantum states the ordinary cohomology (real/complex/de Rham cohomology) H •(X)H^\bullet(X) of the target manifold, but such that the genus=0=0 worldsheet 3-point function (a Gromov-Ruan-Witten invariant) equips the underlying vector space of the cohomology groups with a new product H •(X)⊗H •(X)→H •(X)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 ℂP N−1\mathbb{C}P^{N-1}, N≥2N \geq 2, should be

QH •(ℂP N−1;ℂ)≃ℂ[a 2,b 2N]/(a 2 N−b 2N). 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 •(ℂP N−1;ℂ)≃ℂ[a 2]/(a 2 N). 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)

References

Quantum cohomology rings

The notion of quantum cohomology originates as a model for certain topological string n-point functions in:

motivated by

See also:

The rigorous mathematical formulation in differential symplectic geometry via Gromov-Ruan-Witten invariants is due to:

with early computations in

and in terms of algebraic geometry and via Frobenius manifolds due to:

See also:

More on the history:

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

Introduction and review:

See also:

Relation to chiral algebras:

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

  • Eric Sharpe:

    Quantum sheaf cohomology [pdf] Brandeis university (2010)

    Quantum sheaf cohomology I [pdf]

On the equivariant case:

  • Hiroshi Iritani: Fourier analysis of equivariant quantum cohomology [arXiv:2501.18849]

See also:

  • Anna Hollands, Elad Kosloff, May Sela, Qianyi Shu, Jake P. Solomon: Relative quantum cohomology of the Chiang Lagrangian [arXiv:2305.03016]

Of flag varieties

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

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

Specifically the case of ℂ P N \mathbb{C}P^N ( ℂ P N \mathbb{C}P^N sigma-model):

And specifically the case of ℂ P 1 \mathbb{C}P^1 :

  • Josef F. Dorfmeister, Martin A. Guest, Wayne Rossman, The tt *t t^\ast Structure of the Quantum Cohomology of ℂP 1\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:

and for some form of equivariant K-theory in:

See also:

Relation to D=3 N=2 super Yang-Mills theory:

Relation to D=4 N=1 super Yang-Mills theory:

  • M. Nouman Muteeb, Leopoldo A. Pando Zayas: A Four-dimensional Gauge Theory Perspective on Quantum K-theory [arXiv:2501.02394]

More references are listed in:

In relation to the quantum Yang-Baxter equation:

See also:

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 ℂ P 1 \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 KK 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

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:

On the variant for Pontryagin products not on ordinary homology but in topological K-homology:

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 July 2, 2025 at 06:14:44. See the history of this page for a list of all contributions to it.

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