Landau-Ginzburg models in real mirror symmetry

Walcher, Johannes

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2865-2883.

Récemment, la symétrie miroir pour les cordes ouvertes a dévoilé de nouveaux liens entre la géométrie symplectique et énumérative (modèle A) et la géométrie algébrique complexe (modèle B) qui en un certain sens se situent entre la symétrie miroir classique et sa version homologique. On résume ici le rôle que jouent dans cette histoire les factorisations matricielles et la correspondance Calabi-Yau/Landau-Ginzburg.

In recent years, mirror symmetry for open strings has exhibited some new connections between symplectic and enumerative geometry (A-model) and complex algebraic geometry (B-model) that in a sense lie between classical and homological mirror symmetry. I review the rôle played in this story by matrix factorizations and the Calabi-Yau/Landau-Ginzburg correspondence.

Classification : 81T40, 14N35, 14C25
Keywords: Mirror symmetry, Landau-Ginzburg models, matrix factorizations, algebraic cycles, real enumerative geometry
Mot clés : symétrie miroir, modèle de Landau-Ginzburg, factorisation matricielle, cycle algébrique, géometrie énumérative réelle

[1] Ballard, M.; Favero, D.; Katzarkov, L. A category of kernels for graded matrix factorizations and its implications for Hodge theory (arXiv:1105.3177 [math-AG])

[2] Brunner, I.; Douglas, M. R.; Lawrence, A. E.; Romelsberger, C. D-branes on the quintic, JHEP, Volume 0008 (2000), pp. 015 ([arXiv:hep-th/9906200]) | Zbl

[3] Brunner, I.; Herbst, M.; Lerche, W.; Scheuner, B. Landau-Ginzburg realization of open string TFT (arXiv:hep-th/0305133)

[4] Brunner, I.; Hori, K.; Hosomichi, K.; Walcher, J. Orientifolds of Gepner models, JHEP, Volume 0702 (2007), pp. 001 ([arXiv:hep-th/0401137]) | MR

[5] Buchweitz, R. O. Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings (preprint, ca. 1986)

[6] Candelas, P.; De La Ossa, X. C.; Green, P. S.; Parkes, L. A Pair Of Calabi-Yau Manifolds As An Exactly Soluble Superconformal Theory, Nucl. Phys. B, Volume 359 (1991), pp. 21 | MR | Zbl

[7] Chiodo, A.; Ruan, Y. Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math., Volume 182 (2010), pp. 117-165 | MR | Zbl

[8] Doran, B.; Kirwan, F. Towards non-reductive geometric invariant theory, Pure and applied mathematics quarterly, Volume 3 (2007), pp. 61-105 ([arXiv:math.ag/0703131]) | MR | Zbl

[9] Fan, H.; Jarvis, T. J.; Ruan, Y. The Witten equation and its virtual fundamental cycle (arXiv:0712.4025 [math.AG])

[10] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K. Lagrangian intersection Floer theory—anomaly and obstruction, parts I and II, AMS and International Press, 2009 | Zbl

[11] Givental, A. B. Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices (1996) no. 13, pp. 613-663 ([arXiv:alg-geom/9603021]) | MR | Zbl

[12] Green, M. L. Infinitesimal methods in Hodge theory, Algebraic cycles and Hodge theory (Lecture Notes in Math.), Volume 1594, Springer, 1994, pp. 1-92 | MR | Zbl

[13] Greene, B. R.; Plesser, M. R. Duality in Calabi-Yau moduli space, Nucl. Phys. B, Volume 338 (1990), pp. 15 | MR

[14] Greene, B. R.; Vafa, C.; Warner, N. P. Calabi-Yau Manifolds and Renormalization Group Flows, Nucl. Phys. B, Volume 324 (1989), pp. 371 | MR | Zbl

[15] Herbst, M.; Hori, K.; Page, D. Phases Of N=2 Theories In 1+1 Dimensions With Boundary (arXiv:0803.2045 [hep-th])

[16] Hori, K.; Iqbal, A.; Vafa, C. D-branes and Mirror Symmetry (arXiv:hep-th/0005247)

[17] Hori, K.; Vafa, C. Mirror symmetry (arXiv:hep-th/0002222)

[18] Hori, K.; Walcher, J. D-branes from matrix factorizations. Talk at Strings ’04, June 28–July 2 2004, Paris, Comptes Rendus Physique, Volume 5 (2004), pp. 1061 ([arXiv:hep-th/0409204]) | MR

[19] Hori, K.; Walcher, J. F-term equations near Gepner points, JHEP, Volume 0501 (2005), pp. 008 ([arXiv:hep-th/0404196]) | MR

[20] Hori, K.; Walcher, J. D-brane categories for orientifolds: The Landau-Ginzburg case, JHEP, Volume 0804 (2008), pp. 030 ([arXiv:hep-th/0606179]) | MR | Zbl

[21] Kapustin, A.; Li, Y. D-branes in Landau-Ginzburg models and algebraic geometry, JHEP, Volume 0312 (2003), pp. 005 ([arXiv:hep-th/0210296]) | MR

[22] Kapustin, A.; Li, Y. Topological Correlators in Landau-Ginzburg Models with Boundaries, Adv. Theor. Math. Phys., Volume 7 (2004), pp. 727 ([arXiv:hep-th/0305136]) | MR | Zbl

[23] Kontsevich, M. Enumeration of rational curves via torus actions (arXiv:hep-th/9405035)

[24] Kontsevich, M. Homological algebra of mirror symmetry (Proceedings of I.C.M., (Zürich, 1994)), Birkhäuser, 1995, p. 120-139, [arXiv:math.ag/9411018] | MR | Zbl

[25] Lian, B. .H.; Liu, K.; Yau, S.-T. Mirror Principle I, Surv. Differ. Geom. (1999) no. 5 ([arXiv:alg-geom/9712011]) | MR | Zbl

[26] Mariño, M. Chern-Simons theory, matrix models and topological strings, International Series of Monographs on Physics, 131, Oxford University Press, Oxford, 2005 | MR | Zbl

[27] Morrison, D. R. Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc., Volume 6 (1993), pp. 223-247 ([arXiv:alg-geom/9202004]) | MR | Zbl

[28] Morrison, D. R.; Plesser, M. R. Towards mirror symmetry as duality for two dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl., Volume 46 (1996), pp. 177 ([arXiv:hep-th/9508107]) | MR | Zbl

[29] Morrison, D. R.; Walcher, J. D-branes and Normal Functions, Adv. Theor. Math. Phys., Volume 13 (2009), pp. 553-598 ([arXiv:0709.4028 [hep-th]]) | MR | Zbl

[30] Orlov, D. Derived categories of coherent sheaves and triangulated categories of singularities (arXiv:math.ag/0503632) | Zbl

[31] Orlov, D. Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math., Volume 246 (2004), pp. 227-248 ([arXiv:math.ag/0302304]) | MR | Zbl

[32] Pandharipande, R.; Solomon, J.; Walcher, J. Disk enumeration on the quintic 3-fold, J. Amer. Math. Soc., Volume 21 (2008), pp. 1169-1209 ([arXiv:math.ag/0610901]) | MR | Zbl

[33] Polishchuk, A.; Vaintrob, A. Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (arXiv:1002.2116 [math.AG]) | Zbl

[34] Polishchuk, A.; Vaintrob, A. Matrix factorizations and Cohomological Field Theories (arXiv:1105.2903 [math-AG])

[35] Solomon, J. Intersection Theory on the Moduli Space of Holomorphic Curves with Lagrangian Boundary Conditions, MIT Thesis, 2006 (arXiv:math.sg/0606429) | MR

[36] Takahashi, A. Matrix Factorizations and Representations of Quivers I (arXiv:math.ag/0506347)

[37] van Straten, D. Index theorem for matrix factorizations (Talk at Workshop on Homological Mirror Symmetry and Applications I), Institute for Advanced Study, January, 2007, pp. 22-26

[38] Walcher, J. Residues and Normal Functions (in preparation)

[39] Walcher, J. Stability of Landau-Ginzburg branes, J. Math. Phys., Volume 46 (2005), pp. 082305 ([arXiv:hep-th/0412274]) | MR | Zbl

[40] Walcher, J. Open Strings and Extended Mirror Symmetry (Fields Institute Communications), Volume 54, Proc. BIRS Workshop Modular Forms and String Duality, June 3–8, 2006 | Zbl

[41] Walcher, J. Opening mirror symmetry on the quintic, Comm. Math. Phys., Volume 276 (2007), pp. 671 ([arXiv:hep-th/0605162]) | MR | Zbl

[42] Walcher, J. Extended Holomorphic Anomaly and Loop Amplitudes in Open Topological String, Nucl. Phys. B, Volume 817 (2009), pp. 167 ([arXiv:0705.4098 [hep-th]]) | MR | Zbl

[43] Witten, E. Phases of N = 2 theories in two dimensions, Nucl. Phys. B, Volume 403 (1993), pp. 159 ([arXiv:hep-th/9301042]) | MR | Zbl