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Given a holomorphic vector bundle $E$ over a compact Kähler manifold $X$, one defines twisted Gromov-Witten invariants of $X$ to be intersection numbers in moduli spaces of stable maps $f:\Sigma \to X$ with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle $H^0(\Sigma,f^* E) \ominus H^1(\Sigma,f^* E)$. Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for $X$. This result (Theorem $1$) is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.
@article{10_4007_annals_2007_165_15,
author = {Thomas Coates and Alexander Givental},
title = {Quantum {Riemann{\textendash}Roch,} {Lefschetz} and {Serre}},
journal = {Annals of mathematics},
pages = {15--53},
publisher = {mathdoc},
volume = {165},
number = {1},
year = {2007},
doi = {10.4007/annals.2007.165.15},
mrnumber = {2276766},
zbl = {1189.14063},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.15/}
}
TY - JOUR AU - Thomas Coates AU - Alexander Givental TI - Quantum Riemann–Roch, Lefschetz and Serre JO - Annals of mathematics PY - 2007 SP - 15 EP - 53 VL - 165 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.15/ DO - 10.4007/annals.2007.165.15 LA - en ID - 10_4007_annals_2007_165_15 ER -
%0 Journal Article %A Thomas Coates %A Alexander Givental %T Quantum Riemann–Roch, Lefschetz and Serre %J Annals of mathematics %D 2007 %P 15-53 %V 165 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.15/ %R 10.4007/annals.2007.165.15 %G en %F 10_4007_annals_2007_165_15
Thomas Coates; Alexander Givental. Quantum Riemann–Roch, Lefschetz and Serre. Annals of mathematics, Tome 165 (2007) no. 1, pp. 15-53. doi: 10.4007/annals.2007.165.15
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