On D. Peterson’s comparison formula for Gromov-Witten invariants of 𝐺/𝑃

Woodward, Christopher

doi:10.1090/S0002-9939-05-07709-9

On D. Peterson’s comparison formula for Gromov-Witten invariants of $G/P$
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by Christopher T. Woodward PDF

Proc. Amer. Math. Soc. 133 (2005), 1601-1609 Request permission

Abstract:

We prove a formula of Dale Peterson comparing Gromov-Witten (GW) invariants of $G/P$ to those of $G/B$ using canonical reductions of bundles.

References

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305. MR 1454400, DOI 10.1006/aima.1997.1627
Aaron Bertram, Ionuţ Ciocan-Fontanine, and William Fulton, Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746. MR 1706853, DOI 10.1006/jabr.1999.7960
Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524. MR 1695799, DOI 10.1215/S0012-7094-99-09815-0
Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565–596. MR 1431829, DOI 10.1090/S0894-0347-97-00237-3
W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI 10.1090/pspum/062.2/1492534
W. Fulton and C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), no. 4, 641–661. MR 2072765, DOI 10.1090/S1056-3911-04-00365-0
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138 (French). MR 87176, DOI 10.2307/2372388
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
B. Kim and R. Pandharipande, The connectedness of the moduli space of maps to homogeneous spaces, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 187–201. MR 1882330, DOI 10.1142/9789812799821_{0}006
Andrew Kresch and Harry Tamvakis, Quantum cohomology of orthogonal Grassmannians, Compos. Math. 140 (2004), no. 2, 482–500. MR 2027200, DOI 10.1112/S0010437X03000204
Andrew Kresch and Harry Tamvakis, Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom. 12 (2003), no. 4, 777–810. MR 1993764, DOI 10.1090/S1056-3911-03-00347-3
Augustin-Liviu Mare, Polynomial representatives of Schubert classes in $QH^*(G/B)$, Math. Res. Lett. 9 (2002), no. 5-6, 757–769. MR 1906076, DOI 10.4310/MRL.2002.v9.n6.a5
D. Peterson. Lectures on quantum cohomology of G/P. M.I.T., 1997.
A. Postnikov. Affine approach to quantum Schubert calculus. math.CO/0205165.
A. Ramanathan, Moduli for principal bundles over algebraic curves. I, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 3, 301–328. MR 1420170, DOI 10.1007/BF02867438
C. Teleman and C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 713–748 (English, with English and French summaries). MR 2008438
Jesper Funch Thomsen, Irreducibility of $\overline {M}_{0,n}(G/P,\beta )$, Internat. J. Math. 9 (1998), no. 3, 367–376. MR 1625369, DOI 10.1142/S0129167X98000154