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.
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.
Keywords: (0, 2) mirror symmetry, quantum sheaf cohomology, Landau-Ginzburg model
Mot clés : symétrie miroir (0, 2), cohomologie quantique des faisceaux, modèle Landau-Ginzburg
@article{AIF_2011__61_7_2985_0, author = {Sharpe, Eric}, title = {An introduction to quantum sheaf cohomology}, journal = {Annales de l'Institut Fourier}, pages = {2985--3005}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {7}, year = {2011}, doi = {10.5802/aif.2800}, zbl = {1270.81200}, mrnumber = {3112514}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2800/} }
TY - JOUR AU - Sharpe, Eric TI - An introduction to quantum sheaf cohomology JO - Annales de l'Institut Fourier PY - 2011 SP - 2985 EP - 3005 VL - 61 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2800/ DO - 10.5802/aif.2800 LA - en ID - AIF_2011__61_7_2985_0 ER -
Sharpe, Eric. An introduction to quantum sheaf cohomology. Annales de l'Institut Fourier, Volume 61 (2011) no. 7, pp. 2985-3005. doi : 10.5802/aif.2800. http://www.numdam.org/articles/10.5802/aif.2800/
[1] duality, Adv. Theor. Math. Phys., Volume 7 (2003) no. 5, pp. 865-950 http://projecteuclid.org/getRecord?id=euclid.atmp/1111510433 | MR | Zbl
[2] Topological heterotic rings, Adv. Theor. Math. Phys., Volume 10 (2006) no. 5, pp. 657-682 http://projecteuclid.org/getRecord?id=euclid.atmp/1175791013 | MR | Zbl
[3] Elliptic genera of Landau-Ginzburg models over nontrivial spaces (arXiv: 0905.1285)
[4] mirror symmetry, Nuclear Phys. B, Volume 486 (1997) no. 3, pp. 598-628 | DOI | MR | Zbl
[5] On orbifolds of models, Nuclear Phys. B, Volume 491 (1997) no. 1-2, pp. 263-278 | DOI | MR | Zbl
[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., Volume 58 (2008), pp. 1662-1692 | DOI | MR | Zbl
[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, Volume 338 (1990) no. 1, pp. 15-37 | DOI | MR
[12] Deformed quantum cohomology and mirror symmetry, J. High Energy Phys. (2010) no. 8, pp. 109, 27 | MR
[13] A-twisted heterotic Landau-Ginzburg models, J. Geom. Phys., Volume 59 (2009) no. 12, pp. 1581-1596 | DOI | MR | Zbl
[14] A-twisted Landau-Ginzburg models, J. Geom. Phys., Volume 59 (2009) no. 12, pp. 1547-1580 | DOI | MR | Zbl
[15] Mirror symmetry (hep-th/0002222)
[16] Topological phase of N=2 superconformal field theory and topological Landau-Ginzburg field theory, Phys. Lett., Volume B250 (1990), pp. 91-95 | DOI | MR
[17] Notes on certain correlation functions, Comm. Math. Phys., Volume 262 (2006) no. 3, pp. 611-644 | DOI | MR | Zbl
[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 | Zbl
[19] (0,2) deformations of linear sigma models, J. High Energy Phys. (2011) no. 7, pp. 044 | DOI | MR
[20] Half-twisted correlators from the Coulomb branch, J. High Energy Phys. (2008) no. 4, pp. 071, 19 | DOI | MR | Zbl
[21] Summing the instantons in half-twisted linear sigma models, J. High Energy Phys. (2009) no. 2, pp. 026, 61 | DOI | MR | Zbl
[22] Landau-Ginzburg models and residues, J. High Energy Phys. (2009) no. 9, pp. 118, 25 | DOI | MR
[23] A (0,2) mirror map, J. High Energy Phys. (2011) no. 2, pp. 001 | DOI | MR
[24] Half-twisted Landau-Ginzburg models, J. High Energy Phys. (2008) no. 3, pp. 040, 21 | DOI | MR
[25] Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nuclear Phys. B, Volume 440 (1995) no. 1-2, pp. 279-354 | DOI | MR | Zbl
[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 | Zbl
[27] Notes on correlation functions in theories, Snowbird lectures on string geometry (Contemp. Math.), Volume 401, Amer. Math. Soc., Providence, RI, 2006, pp. 93-104 | MR | Zbl
[28] Notes on certain other correlation functions, Adv. Theor. Math. Phys., Volume 13 (2009) no. 1, pp. 33-70 http://projecteuclid.org/getRecord?id=euclid.atmp/1232551519 | MR | Zbl
[29] Topological Landau-Ginzburg models, Mod. Phys. Lett., Volume A6 (1991), pp. 337-346 | DOI | MR | Zbl
[30] Mirror manifolds and topological field theory, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 120-158 | MR | Zbl
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