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stub for quantum sheaf cohomology
Added canonical references:
After the original physics ideas of Vafa and Witten, a differential geometric formalization was pioneered in
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
found the DOI-s for Ruan & Tian and copied the items to their author-pages
added the original references
Cumrun Vafa, 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, Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry 1 (1990) [doi:10.4310/SDG.1990.v1.n1.a5, inspire:307956]
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]
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”.
added pointers to:
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]
Josef F. Dorfmeister, Martin A. Guest, Wayne Rossman , The $tt*$ 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]
added statement (here) of the quantum cohomology ring of $\mathbb{C}P^N$, with a bunch of the early physics references.
Its curious that the expression
$QH^\bullet\big( \mathbb{C}P^{N-1} ;\, \mathbb{C} \big) \;\simeq\; \mathbb{C}\big[ a_2,\, b_{2N} \big]/(a_2^N - b_{2N}) \,.$looks like it wants to be the “higher universal enveloping algebra” of the Whitehead-bracket $L_\infty$-algebra of $\mathbb{C}P^{N-1}$ (by this formula).
I was thinking that, at least for $N = 2$, 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.
added pointer to today’s
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