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nLab > Latest Changes: quantum cohomology ring

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  1.  
    • 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 ℂP N\mathbb{C}P^N, with a bunch of the early physics references.

    Its curious that the expression

    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}) \,.

    looks like it wants to be the “higher universal enveloping algebra” of the Whitehead-bracket L ∞L_\infty-algebra of ℂP N−1\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

1 to 12 of 12

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