Dans ces notes nous passons en revue la « cohomologie quantique des faisceaux », une déformation de la cohomologie des faisceaux qui apparaît d’une façon similaire à la cohomologie quantique ordinaire (tout en la généralisant parfois). La cohomologie quantique des faisceaux apparaît dans l’étude de la symétrie miroir (0,2), ce qui est passé en revue. Après ça nous passons en revue la théorie standard des champs topologique et les modèles A/2, B/2, dans lesquels la cohomologie quantique des faisceaux apparaît, et esquissons les définitions basiques et les calculs. Ensuite nous discutons dans ce contexte les modèles de supersymétrie Landau-Ginzburg (2,2) et (0,2) ainsi que la cohomologie quantique des faisceaux.
In this note we review “quantum sheaf cohomology,” a deformation of sheaf cohomology that arises in a fashion closely akin to (and sometimes generalizing) ordinary quantum cohomology. Quantum sheaf cohomology arises in the study of (0,2) mirror symmetry, which we review. We then review standard topological field theories and the A/2, B/2 models, in which quantum sheaf cohomology arises, and outline basic definitions and computations. We then discuss (2,2) and (0,2) supersymmetric Landau-Ginzburg models, and quantum sheaf cohomology in that context.
@article{AIF_2011__61_7_2985_0,
author = {Sharpe, Eric},
title = {An introduction to quantum sheaf cohomology},
journal = {Annales de l'Institut Fourier},
volume = {61},
year = {2011},
pages = {2985-3005},
doi = {10.5802/aif.2800},
zbl = {1270.81200},
mrnumber = {3112514},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2011__61_7_2985_0}
}
Sharpe, Eric. An introduction to quantum sheaf cohomology. Annales de l'Institut Fourier, Tome 61 (2011) pp. 2985-3005. doi : 10.5802/aif.2800. http://gdmltest.u-ga.fr/item/AIF_2011__61_7_2985_0/
[1] duality, Adv. Theor. Math. Phys., Tome 7 (2003) no. 5, pp. 865-950 http://projecteuclid.org/getRecord?id=euclid.atmp/1111510433 | MR 2045304 | Zbl 1058.81064
[2] Topological heterotic rings, Adv. Theor. Math. Phys., Tome 10 (2006) no. 5, pp. 657-682 http://projecteuclid.org/getRecord?id=euclid.atmp/1175791013 | MR 2281544 | Zbl 1116.81049
[3] Elliptic genera of Landau-Ginzburg models over nontrivial spaces (arXiv: 0905.1285)
[4] mirror symmetry, Nuclear Phys. B, Tome 486 (1997) no. 3, pp. 598-628 | Article | MR 1436158 | Zbl 0925.14014
[5] On orbifolds of models, Nuclear Phys. B, Tome 491 (1997) no. 1-2, pp. 263-278 | Article | MR 1449061 | Zbl 0925.32013
[6] A mathematical theory of quantum sheaf cohomology (arXiv: 1110.3751)
[7] Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varieties (arXiv: 1110.3752)
[8] GLSM’s for partial flag manifolds, J. Geom. Phys., Tome 58 (2008), pp. 1662-1692 | Article | MR 2468445 | Zbl 1218.81091
[9] The Witten equation and its virtual fundamental cycle (arXiv: 0712.4025)
[10] The Witten equation, mirror symmetry, and quantum singularity theory (arXiv: 0712.4021)
[11] Duality in Calabi-Yau moduli space, Nuclear Phys. B, Tome 338 (1990) no. 1, pp. 15-37 | Article | MR 1059831
[12] Deformed quantum cohomology and mirror symmetry, J. High Energy Phys. (2010) no. 8, pp. 109, 27 | MR 2756043
[13] A-twisted heterotic Landau-Ginzburg models, J. Geom. Phys., Tome 59 (2009) no. 12, pp. 1581-1596 | Article | MR 2583794 | Zbl 1187.81189
[14] A-twisted Landau-Ginzburg models, J. Geom. Phys., Tome 59 (2009) no. 12, pp. 1547-1580 | Article | MR 2583793 | Zbl 1187.81188
[15] Mirror symmetry (hep-th/0002222)
[16] Topological phase of N=2 superconformal field theory and topological Landau-Ginzburg field theory, Phys. Lett., Tome B250 (1990), pp. 91-95 | Article | MR 1081501
[17] Notes on certain correlation functions, Comm. Math. Phys., Tome 262 (2006) no. 3, pp. 611-644 | Article | MR 2202305 | Zbl 1109.81066
[18] Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel (1995), pp. 120-139 | MR 1403918 | Zbl 0846.53021
[19] (0,2) deformations of linear sigma models, J. High Energy Phys. (2011) no. 7, pp. 044 | Article | MR 2875974
[20] Half-twisted correlators from the Coulomb branch, J. High Energy Phys. (2008) no. 4, pp. 071, 19 | Article | MR 2425232 | Zbl 1246.81339
[21] Summing the instantons in half-twisted linear sigma models, J. High Energy Phys. (2009) no. 2, pp. 026, 61 | Article | MR 2486403 | Zbl 1245.81249
[22] Landau-Ginzburg models and residues, J. High Energy Phys. (2009) no. 9, pp. 118, 25 | Article | MR 2580690
[23] A (0,2) mirror map, J. High Energy Phys. (2011) no. 2, pp. 001 | Article | MR 2820839
[24] Half-twisted Landau-Ginzburg models, J. High Energy Phys. (2008) no. 3, pp. 040, 21 | Article | MR 2391080
[25] Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nuclear Phys. B, Tome 440 (1995) no. 1-2, pp. 279-354 | Article | MR 1336089 | Zbl 0908.14014
[26] Towards mirror symmetry as duality for two-dimensional abelian gauge theories, Strings ’95 (Los Angeles, CA, 1995), World Sci. Publ., River Edge, NJ (1996), pp. 374-387 | MR 1660724 | Zbl 0957.81656
[27] Notes on correlation functions in theories, Snowbird lectures on string geometry, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 401 (2006), pp. 93-104 | MR 2222532 | Zbl 1172.14331
[28] Notes on certain other correlation functions, Adv. Theor. Math. Phys., Tome 13 (2009) no. 1, pp. 33-70 http://projecteuclid.org/getRecord?id=euclid.atmp/1232551519 | MR 2471852 | Zbl 1171.81420
[29] Topological Landau-Ginzburg models, Mod. Phys. Lett., Tome A6 (1991), pp. 337-346 | Article | MR 1093562 | Zbl 1020.81886
[30] Mirror manifolds and topological field theory, Essays on mirror manifolds, Int. Press, Hong Kong (1992), pp. 120-158 | MR 1191422 | Zbl 0904.58009