Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2011
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2023

    Added canonical references:

    After the original physics ideas of Vafa and Witten, a differential geometric formalization was pioneered in

    • Yongbin Ruan, Gang Tian, A mathematical theory of quantum cohomology, Mathematical Research Letters 1 (1994) 269-278
    • Yongbin Ruan, Gang Tian, A mathematical theory of quantum cohomology, J. Diff. Geometry 42:2 (1995) 259-367

    and in the algebraic geometric terms by Manin and Kontsevich,

    Somewhat equivalent approach by Frobenius manifolds has been independently pushed by Dubrovin with motivation in integrable systems.

    A comprehensive early monograph is

    • Yuri Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloqium Publications 47, 1999

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2023

    found the DOI-s for Ruan & Tian and copied the items to their author-pages

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023
    • (edited Jul 5th 2023)

    added this pointer:

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    Have !include-ed (here) a list of references on the relation between quantum cohomology of flag manifolds and Pontrjagin rings

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    added the original references

    and added more introductory references:

    • Martin 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]

    • Tom Coates, An Introduction to Quantum Cohomology [pdf]

    • Alexander Givental, A tutorial on Quantum Cohomology [pdf]

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2023

    I have rewritten and somewhat expanded the little bit of Idea-section text that we had here, meaning to make it more to the point (but it remains minimalistic).

    In particular I added a warning paragraph that it is not the notion of cohomology but of the cup/wedge-product ring structure that is being deformed/quantized here, whence the original and appropriate terminology is quantum cohomology ring instead of just quantum cohomology.

    Finally, in this vein, I am renaming the entry from “quantum sheaf cohomology” to “quantum cohomology ring”.

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2023

    added pointers to:

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2023
    • (edited Jul 9th 2023)

    added statement (here) of the quantum cohomology ring of P N\mathbb{C}P^N, with a bunch of the early physics references.

    Its curious that the expression

    QH (P N1;)[a 2,b 2N]/(a 2 Nb 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}) \,.

    looks like it wants to be the “higher universal enveloping algebra” of the Whitehead-bracket L L_\infty-algebra of P N1\mathbb{C}P^{N-1} (by this formula).

    I was thinking that, at least for N=2N = 2, this should hence be an example of the relation to the quantum cohomology to the Pontrjagin ring (here) combined with the relation of the Pontrjagin ring to the universal envelope of the Whitehead bracket (here) – but when I try to write this out it fails by some dimension shifts.

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2023

    started adding (here) references on quantum K-theory rings

    diff, v16, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 14th 2023
    • (edited Sep 14th 2023)

    added pointer to today’s

    • Cyril Closset, Osama Khlaif, Grothendieck lines in 3d 𝒩=2\mathcal{N}=2 SQCD and the quantum K-theory of the Grassmannian [arXiv:2309.06980]

    diff, v18, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2023
    • (edited Nov 26th 2023)

    added a line to the example (here) making more explicit how the quantum cohomology ring is a deformation of the ordinary cohomology ring.

    diff, v22, current