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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].

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
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References:

[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
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