nLab quantum cohomology as Pontrjagin rings -- references

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$]$