A brief survey of FJRW theory. (English) Zbl 1452.14056
Hori, Kentaro (ed.) et al., Primitive forms and related subjects – Kavli IPMU 2014. Proceedings of the international conference, University of Tokyo, Tokyo, Japan, February 10–14, 2014. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 83, 19-53 (2019).
Summary: In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field, including the gauged linear sigma model, which is expected to provide a geometric framework for unifying many of these ideas.
For the entire collection see [Zbl 1446.53004].
For the entire collection see [Zbl 1446.53004].
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 32S05 | Local complex singularities |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |
Keywords:
FJRW theory; GLSM; gauged linear sigma model; Gromov-Witten; quantum singularity; Saito-Givental; Landau-Ginzburg; Calabi-YauReferences:
| [1] | P. Acosta. Asymptotic Expansion and the LG/(Fano, General Type) Correspondence.ArXiv e-prints, November 2014, 1411.4162. |
| [2] | A. Buryak, B. Dubrovin, J. Gu´er´e and P. Rossi. Tau-structure for the Double Ramification Hierarchies. Communications in Mathematical Physics, 363(1) 191-260, 2018. · Zbl 1431.53095 |
| [3] | A. Buryak and J. Gu´er´e. Towards a description of the double ramification hierarchy for Witten’sr-spin class.Journal de Math´ematiques Pures et Appliqu´ees, 2016. · Zbl 1352.37176 |
| [4] | A. Basalaev and N. Priddis. Givental-type reconstruction at a nonsemisimple point.Michigan Math. J., 67(2):333-369, 2018. · Zbl 1441.14135 |
| [5] | A. Buryak, H. Posthuma and S. Shadrin. A polynomial bracket for the Dubrovin-Zhang hierarchies.J. Differential Geom., 92(1):153-185, 2012. · Zbl 1259.53079 |
| [6] | A. Buryak and P. Rossi. Double ramification cycles and quantum integrable systems.Lett. Math. Phys., 106(3):289-317, 2016. · Zbl 1338.14030 |
| [7] | A. Basalaev and A. Takahashi. On rational Frobenius manifolds of rank three with symmetries.Journal of Geometry and Physics, 84:73-86, October 2014, 1401.3505. · Zbl 1417.53096 |
| [8] | D. Cheong, I. Ciocan-Fontanine and B. Kim. Orbifold quasimap theory.Math. Ann., 363(3-4):777-816, 2015. · Zbl 1337.14012 |
| [9] | P. Candelas, X. C. de la Ossa, P. S. Green and L. Parkes. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. InEssays on mirror manifolds, pages 31-95. Int. Press, Hong Kong, 1992. · Zbl 0826.32016 |
| [10] | I. Ciocan-Fontanine and B. Kim. Moduli stacks of stable toric quasimaps.Adv. Math., 225(6):3022-3051, 2010. · Zbl 1203.14014 |
| [11] | I. Ciocan-Fontanine, B. Kim and D. Maulik. Stable quasimaps to GIT quotients.J. Geom. Phys., 75:17-47, 2014. · Zbl 1282.14022 |
| [12] | A. Chiodo. A construction of Witten’s top Chern class inK-theory. InGromov-Witten theory of spin curves and orbifolds, volume |
| [13] | A. Chiodo. The Witten top Chern class viaK-theory.J. Algebraic Geom., 15(4):681-707, 2006. · Zbl 1117.14008 |
| [14] | A. Chiodo, H. Iritani and Y. Ruan. Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence. Publ. Math. Inst. Hautes ´Etudes Sci., 119:127-216, 2014. · Zbl 1298.14042 |
| [15] | H.-L. Chang and J. Li. Gromov-Witten invariants of stable maps with fields.Int. Math. Res. Not. IMRN, (18):4163-4217, 2012. · Zbl 1253.14053 |
| [16] | E. Clader. Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X{3,3}and X{2,2,2,2}.Adv. Math., 307:1-52, 2017. · Zbl 1356.14048 |
| [17] | H.-L. Chang, J. Li and W.-P. Li. Witten’s top Chern class via cosection localization.Invent. Math., 200(3):1015-1063, 2015. · Zbl 1318.14048 |
| [18] | A. Chiodo and Y. Ruan. Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations.Invent. Math., 182(1):117-165, 2010. · Zbl 1197.14043 |
| [19] | A. Chiodo and Y. Ruan. A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence.Ann. Inst. Fourier(Grenoble), 61(7):2803-2864, 2011. · Zbl 1408.14125 |
| [20] | A. Chiodo and Y. Ruan. LG/CY correspondence: the state space isomorphism.Adv. Math., 227(6):2157-2188, 2011. · Zbl 1245.14038 |
| [21] | V. G. Drinfeld and V. V. Sokolov. Lie algebras and equations of Korteweg-de Vries type. InCurrent problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, pages 81-180. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984. · Zbl 0558.58027 |
| [22] | B. Dubrovin and Y. Zhang. Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants.ArXiv Mathematics e-prints, August 2001, math/0108160. |
| [23] | E. Frenkel, A. Givental and T. Milanov. Soliton equations, vertex operators, and simple singularities.Funct. Anal. Other Math., 3(1):47-63, 2010. · Zbl 1203.37108 |
| [24] | A. Francis, T. Jarvis, D. Johnson and R. Suggs. Landau-Ginzburg mirror symmetry for orbifolded Frobenius algebras. InStringMath 2011, volume 85 ofProc. Sympos. Pure Math., pages 333- 353. Amer. Math. Soc., Providence, RI, 2012. |
| [25] | H. Fan, T. J. Jarvis and Y. Ruan. The Witten equation and its virtual fundamental cycle.ArXiv e-prints, December 2007, 0712.4025. |
| [26] | H. Fan, T. Jarvis, and Y. Ruan. The Witten equation, mirror symmetry, and quantum singularity theory.Ann. of Math.(2), 178(1):1-106, 2013. · Zbl 1310.32032 |
| [27] | H. Fan, T. Jarvis, and Y. Ruan. The analytic theory of gauged Witten equation and its virtual fundamental cycle.In preparation, 2016. |
| [28] | H. Fan, T. Jarvis and Y. Ruan. A Mathematical Theory of the Gauged Linear Sigma Model.Geom. Topol., 22(1):235-303, 2018. · Zbl 1388.14041 |
| [29] | A. Francis. Computational techniques in FJRW theory with applications to Landau-Ginzburg mirror symmetry.Adv. Theor. Math. Phys., 19(6):1339-1383, 2015. · Zbl 1342.81515 |
| [30] | C. Faber, S. Shadrin, and D. Zvonkine. Tautological relations and ther-spin Witten conjecture.Ann. Sci. ´Ec. Norm. Sup´er.(4), 43(4):621-658, 2010. · Zbl 1203.53090 |
| [31] | A. B. Givental. Equivariant Gromov-Witten invariants.Internat. Math. Res. Notices, (13):613-663, 1996. · Zbl 0881.55006 |
| [32] | A. B. Givental. Gromov-Witten invariants and quantization of quadratic Hamiltonians.Mosc. Math. J., 1(4):551-568, 645, 2001. Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. · Zbl 1008.53072 |
| [33] | A. Givental.An−1singularities andnKdV hierarchies.Mosc. Math. J., 3(2):475-505, 743, 2003. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. · Zbl 1054.14067 |
| [34] | A. B. Givental. Symplectic geometry of Frobenius structures. In Frobenius manifolds, Aspects Math., E36, pages 91-112. Friedr. Vieweg, Wiesbaden, 2004. · Zbl 1075.53091 |
| [35] | A. B. Givental and T. E. Milanov. Simple singularities and integrable hierarchies. InThe breadth of symplectic and Poisson geometry, volume 232 ofProgr. Math., pages 173-201. Birkh¨auser Boston, Boston, MA, 2005. |
| [36] | B. Greene and P. Ronen. Duality in Calabi-Yau moduli space. Nuclear Physics B., 338(1):15-37, 1990. |
| [37] | W. He, S. Li, Y. Shen and R. Webb. Landau-Ginzburg Mirror Symmetry Conjecture.ArXiv e-prints, March 2015, 1503.01757. |
| [38] | T. Hollowood and J. L. Miramontes. Tau-functions and generalized integrable hierarchies.Comm. Math. Phys., 157(1):99-117, 1993. · Zbl 0796.35144 |
| [39] | K. Intriligator and C. Vafa. Landau-ginzburg orbifolds.Nuclear Physics B, 339(1):95-120, 1990. |
| [40] | F. Janda. Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures. ArXiv e-prints, July 2014, 1407.4778. |
| [41] | F. Janda. Relations in the Tautological Ring and Frobenius Manifolds near the Discriminant.ArXiv e-prints, May 2015, 1505.03419. |
| [42] | T. J. Jarvis, T. Kimura and A. Vaintrob. Moduli spaces of higher spin curves and integrable hierarchies.Compositio Mathematica, 126:157, 2001. · Zbl 1015.14028 |
| [43] | R. M. Kaufmann. Orbifold frobenius algebras, cobordisms and monodromies. In D. DeTurck, editor,Contemporary Mathematics, pages 135-161, Providence, R.I, 2002. American Mathematical Society. · Zbl 1084.57027 |
| [44] | R. M. Kaufmann. Orbifolding Frobenius algebras.Internat. J. Math., 14(6):573-617, 2003. · Zbl 1083.57037 |
| [45] | R. M. Kaufmann. Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry.Gromov-Witten theory of spin curves and orbifolds, volume 403 ofContemp. Math., pages 67-116. Amer. Math. Soc., Providence, RI, 2006. · Zbl 1116.14037 |
| [46] | B. Kim. Stable quasimaps.Commun. Korean Math. Soc., 27(3):571-581, 2012. · Zbl 1245.14058 |
| [47] | Y.-H. Kiem and J. Li. Localizing virtual cycles by cosections.J. Amer. Math. Soc., 26(4):1025-1050, 2013. · Zbl 1276.14083 |
| [48] | M. Kontsevich and Y. Manin. Gromov-Witten classes, quantum cohomology, and enumerative geometry.Comm. Math. Phys., 164(3):525-562, 1994. · Zbl 0853.14020 |
| [49] | M. Krawitz.FJRW Rings and Landau-Ginzburg Mirror Symmetry. PhD thesis, The University of Michigan, 2010. · Zbl 1250.81087 |
| [50] | M. Kreuzer and H. Skarke. On the classification of quasihomogeneous functions.Communications in Mathematical Physics, 150:137-147, November 1992, arXiv:hep-th/9202039. · Zbl 0767.57019 |
| [51] | M. Krawitz and Y. Shen. Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic OrbifoldP1.ArXiv e-prints, June 2011, 1106.6270. |
| [52] | V. G. Kac and M. Wakimoto. Exceptional hierarchies of soliton equations. InTheta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), volume 49 ofProc. Sympos. Pure Math., pages 191-237. Amer. Math. Soc., Providence, RI, 1989. · Zbl 0691.17014 |
| [53] | Y.-P. Lee. Witten’s conjecture and the Virasoro conjecture for genus up to two. InGromov-Witten theory of spin curves and orbifolds, volume 403 ofContemp. Math., pages 31-42. Amer. Math. Soc., Providence, RI, 2006. · Zbl 1114.14034 |
| [54] | B. Lee.G-Frobenius Manifolds.J. Algebra, 479:35-77, 2017. · Zbl 1388.53097 |
| [55] | C. Li, S. Li, K. Saito and Y. Shen. Mirror symmetry for exceptional unimodular singularities.J. Eur. Math. Soc.(JEMS), 19(4):1189-1229, 2017. · Zbl 1387.14021 |
| [56] | B. H. Lian, K. Liu and S.-T. Yau. Mirror principle. I [MR1621573 (99e:14062)]. InSurveys in differential geometry: differential |
| [57] | Y.-P. Lee, N. Priddis and M. Shoemaker. A proof of the LandauGinzburg/Calabi-Yau correspondence via the crepant transformation conjecture.Ann. Sci. ´Ec. Norm. Sup´er.(4), 49(6):1403- 1443, 2016. · Zbl 1360.14133 |
| [58] | Y.-P. Lee and M. Shoemaker. A mirror theorem for the mirror quintic.Geom. Topol., 18(3):1437-1483, 2014. · Zbl 1305.14025 |
| [59] | S.-Q. Liu, C.-Z. Wu and Y. Zhang. On the Drinfeld-Sokolov hierarchies ofDtype.Int. Math. Res. Not. IMRN, (8):1952-1996, 2011. · Zbl 1221.35458 |
| [60] | E. J. Martinec. Criticality, catastrophes, and compactifications. In Physics and mathematics of strings, pages 389-433. World Sci. Publ., Teaneck, NJ, 1990. · Zbl 0737.58060 |
| [61] | T. Mochizuki. The virtual class of the moduli stack of stabler-spin curves.Comm. Math. Phys., 264(1):1-40, 2006. · Zbl 1136.14015 |
| [62] | A. Marian, D. Oprea and R. Pandharipande. The moduli space of stable quotients.Geom. Topol., 15(3):1651-1706, 2011. · Zbl 1256.14057 |
| [63] | T. Milanov and Y. Ruan. Gromov-Witten theory of elliptic orbifoldP1and quasi-modular forms.ArXiv e-prints, June 2011, 1106.2321. |
| [64] | T. Milanov and Y. Shen. Global mirror symmetry for invertible simple elliptic singularities.Ann. Inst. Fourier(Grenoble), 66(1):271-330, 2016. · Zbl 1366.14039 |
| [65] | R. Pandharipande, A. Pixton and D. Zvonkine. Relations onMg,n via 3-spin structures.J. Amer. Math. Soc., 28(1):279-309, 2015. · Zbl 1315.14037 |
| [66] | R. Pandharipande, A. Pixton and D. Zvonkine. Tautological relations via r-spin structures.ArXiv e-prints, July 2016, 1607.00978. · Zbl 1433.14022 |
| [67] | N. Priddis and M. Shoemaker. A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic.Ann. Inst. Fourier (Grenoble), 66(3):1045-1091, 2016. · Zbl 1354.14082 |
| [68] | A. Polishchuk and A. Vaintrob. Algebraic construction of Witten’s top Chern class. InAdvances in algebraic geometry motivated by physics(Lowell, MA, 2000), volume 276 ofContemp. Math., pages 229-249. Amer. Math. Soc., Providence, RI, 2001. · Zbl 1051.14007 |
| [69] | A. Polishchuk and A. Vaintrob. Matrix factorizations and cohomological field theories.J. Reine Angew. Math., 714:1-122, 2016. · Zbl 1357.14024 |
| [70] | K. Saito. Primitive forms for a universal unfolding of a function with an isolated critical point.J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28(3):775-792 (1982), 1981. · Zbl 0523.32015 |
| [71] | K. Saito. The higher residue pairingsKF(k)for a family of hypersurface singular points. InSingularities, Part 2(Arcata, Calif., 1981), volume 40 ofProc. Sympos. Pure Math., pages 441-463. Amer. Math. Soc., Providence, RI, 1983. · Zbl 0565.32005 |
| [72] | K. Saito. Period mapping associated to a primitive form.Publ. Res. Inst. Math. Sci., 19(3):1231-1264, 1983. · Zbl 0539.58003 |
| [73] | K. Saito. From primitive form to mirror symmetry.ArXiv e-prints, August 2014, 1408.4208. |
| [74] | K. Saito and A. Takahashi. From primitive forms to Frobenius manifolds. InFrom Hodge theory to integrability and TQFT tt*geometry, volume 78ofProc. Sympos. Pure Math., pages 31-48. Amer. Math. Soc., Providence, RI, 2008. · Zbl 1161.32013 |
| [75] | I. Satake and A. Takahashi. Gromov-Witten invariants for mirror orbifolds of simple elliptic singularities.Ann. Inst. Fourier (Grenoble), 61(7):2885-2907, 2011. · Zbl 1294.14016 |
| [76] | J. Tay.Poincar´e Polynomial of FJRW Rings and the Group-weights Conjecture. Brigham Young University. Department of Mathematics, 2013. |
| [77] | C. Teleman. The structure of 2D semi-simple field theories.Invent. Math., 188(3):525-588, 2012. · Zbl 1248.53074 |
| [78] | G. Tian and G. Xu. Analysis of gauged Witten equation.J. Reine Angew. Math., 740:187-274, 2018. · Zbl 1411.32017 |
| [79] | C. Vafa and N. Warner. Catastrophes and the classification of conformal theories.Physics Letters B, 218:51-58, February 1989. |
| [80] | C. T. C. Wall. A note on symmetry of singularities.Bulletin of the London Mathematical Society, 12(3):169-175, 1980, http://blms.oxfordjournals.org/content/12/3/169.full.pdf+html. · Zbl 0427.32010 |
| [81] | C. T. C. Wall. A second note on symmetry of singularities.Bulletin of the London Mathematical Society, 12(5):347-354, 1980, http://blms.oxfordjournals.org/content/12/5/347.full.pdf+html. · Zbl 0424.58006 |
| [82] | E. Witten. TheN-matrix model and gauged WZW models.Nuclear Physics B, 371:191-245, March 1992. |
| [83] | E. Witten. Algebraic geometry associated with matrix models of two dimensional gravity. InTopological methods in modern mathematics(Stony Brook, NY, 1991), pages 235-269. Publish or Perish, Houston, TX, 1993. · Zbl 0812.14017 |
| [84] | E. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
